trigonometry final exam review, spring 2010faculty.madisoncollege.edu/kmirus/20092010b/804213… ·...

Trigonometry FINAL EXAM Review Sheet of 52

Section 1.1: Angles Complementary and Supplementary Angles

Complementary angles have measures that add up to 90. Supplementary angles have measures that add up to 180.

Fractions of a Degree

One minute of angle is one-sixtieth of a degree:

One second of angle is one-sixtieth of a minute:

Example of an angle measurement stated in degrees, minutes, and seconds (DMS):

Converting from degrees, minutes, and seconds (DMS) to decimal degrees (DD) Example: convert to DD Divide the number of second by sixty to convert it to an equivalent number in minutes.

o

Add that to the number of minutes, then divide by sixty again to convert to an equivalent number in degrees.

o

Add that number to the number of degrees, and round to an appropriate number of places:o

Note that . Thus, stating DMS measurements to the nearest

thousandth of a degree won’t result in much round-off error, but stating your answer beyond a ten-thousandth of a degree implies a precision not conveyed by mere seconds.

Note: on a graphing calculator, you can type in a DMS measurement using the degree and minute symbols found in the ANGLE menu, and the quotation marks for the seconds.

Converting from decimal degrees (DD) to degrees, minutes, and seconds (DMS) Example: convert to DMS Multiply the decimal portion by sixty to convert it to an equivalent number in minutes.

o

Multiply the resulting decimal portion by sixty again to convert to an equivalent number in seconds.

o

Round to an appropriate number of places:o


Note: on a graphing calculator, you can type in a DD measurement and convert it to a DMS measurement using the DMS function found in the ANGLE menu.

Standard Position An angle in standard position has its vertex at the origin of a Cartesian coordinate system

and its initial side on the positive x-axis. Thus, acute angles have their terminal sides in Quadrant I. Also, obtuse angles have their terminal sides in Quadrant II. A quadrantal angle has its terminal side on the x- or y-axis. Quadrantal angles have measure of 90, 180, 270, 360, 450, 540, 630, 720, …, -90, -

180, -270, -360,-450, etc.

Coterminal Angles Coterminal angles have the same terminal side, but differ in how many rotations around

the circle were taken to get to that side. The measures of coterminal angles differ by a multiple of 360. Angles coterminal with a given angle measure of will have measures given by

+ (360)n for any integer n.

Section 1.2: Angle Relationships and Similar Triangles

Geometric Properties of AnglesVertical angles are on opposite sides of the intersection point of two lines.

Vertical angels have equal measures.Parallel lines lie in the same plane and do not intersect.A transversal is a line that intersects two parallel lines.

Different angles formed by the transversal are given special names to reflect special relationships between their measures:

o Corresponding angles have equal measures (this is a postulate (i.e., a fact that is accepted without proof) which is used to prove the following relationships).

o Alternate interior angles have equal measures.o Alternate exterior angles have equal measures.o Interior angles on the same side of the transversal are supplementary (add to

180).


TrianglesThe sum of the measures of the angles of any triangle is 180.

Types of TrianglesTriangles are classified according to their angles and sides.

Angle classifications:o Acute triangles have all angles less than 90.o Right triangles have one angle of 90.o Obtuse triangles have one angle greater than 90.

Side classifications:o Equilateral triangles have all sides the same length.o Isosceles triangles have two sides of the same length.o Scalene triangles have no sides of the same length.

Congruent triangles have all the same side lengths and all the same angles. Congruent is the proper way to say that two triangles are “equal” or “the same”.Similar triangles have the same shape (i.e., all the same angles), but not necessarily the same size. The larger triangle is like a “magnification” of the smaller one.

Conditions for Similar Triangles:If triangle ABC is similar to triangle DEF, then the following conditions must hold:

Corresponding angles must have the same measure (a postulate). Corresponding sides must be proportional.


Section 1.3: Trigonometric Functions

The Six Trigonometric Functions are defined as follows:

(sine) (cosine)

(tangent) (cotangent)

(secant) (cosecant)


Section 1.4: Using the Definitions of the Trigonometric Functions

The Reciprocal Identities (Section 1.4)

Note: we can re-state these reciprocal identities as:

Signs and Ranges of the Trigonometric FunctionsCombining the signs of x and y in the four quadrants with the definitions of the trig functions allows us to state the sign of the trig functions for any angle in a given quadrant.

Range of the Six Trigonometric FunctionsTrig Function Range stated in interval

notationRange stated in set builder notation

The Pythagorean Identities (Section 1.4)

The Quotient Identities (Section 1.4)


Section 2.1: Trigonometric Functions of Acute Angles

Right Triangle Based Definitions of Trigonometric Functions (Section 2.1)(SOH CAH TOA)

Cofunction Identities (Section 2.1)For any acute angle A,


Trigonometric Function Values for 45 (Section 2.1)


Trigonometric Function Values for 30 and 60 (Section 2.1)


Section 2.2: Trigonometric Functions of Non-Acute AnglesReference AnglesA reference angle is the angle formed between the terminal side of any angle and the x axis.Notice that the reference angle is always an acute angle. This means that the trig function values of acute angles can be used to compute the trig function values of any angle. All you have to do is get the signs right for x and y based on the quadrant of the terminal side.

Reference angle for terminal side in Quadrant II

Reference angle for terminal side in Quadrant III

Reference angle for terminal side in Quadrant IV


Section 2.3: Finding Trigonometric Function Values Using a CalculatorEvaluating Sine, Cosine, and Tangent on a Calculator

Make sure your calculator is in degree mode.

Enter the trig function followed by the angle in parentheses.

For angles in DMS, you can enter the angle using the DMS functionality of your calculator.

For angles in DMS, you also can enter the angle as the number of degrees plus fractions of a degree.


Evaluating Secant, Cosecant, and Cotangent on a Calculator Calculators do not have buttons for these functions. You have to use the reciprocal identities to evaluate these trig functions on a calculator.

The Reciprocal Identities (Section 1.4)

To enter the reciprocal calculations correctly, you have to enter 1 divided by the correct trig function. DO NOT use the SIN-1, COS-1, or TAN-1 buttons for cosecant, secant, or cotangent; those are the inverse functions (not the reciprocals).

Finding Angle Measures on a Calculator We can use this inverse function property notion to solve equations where a trig function

of an unknown angle is equal to a constant. We then just take the inverse function of both sides of the equation to find the angle.o The inverse sine function is written as sin-1.o The inverse cosine function is written as cos-1.o The inverse tangent function is written as tan-1.

Note that when you use the inverse trig functions on a calculator, o The sin-1 function returns an angle in the interval .o The cos-1 function returns an angle in the interval .o The tan-1 function returns an angle in the interval .


To find an angle for the secant, cosecant, and cotangent functions, you have to use the reciprocal identities first to convert the equation so that it has cosine, sine, or tangent.


Section 2.4: Solving Right Triangles The way we write a measurement can be used to indicate how precise the measurement was.

o If we write 15 inches, that means the object we measured was between 14.5 and 15.5 inches long.

o If we write 15.0 inches, that means the object we measured was between 14.95 and 15.05 inches long.

o If we write 15.00 inches, that means the object we measured was between 14.995 and 15.005 inches long.

o In other words, a measurement is good to ½ of the least significant digit.This is called using significant digits to write a number.When using trig functions, the rules for writing significant digits are as follows:

Angles and Accuracy of Trig FunctionsAngle Precision Trig Function Accuracy Example

1 2 sig. figs. , 0.1 or 10 3 sig. figs. ,

0.01 or 1 4 sig. figs. ,

0.001 or 10 5 sig. figs. ,

To solve a right triangle:1. Make a sketch of the triangle, label sides and angles consistently (a, b, and c for the legs

and hypotenuse; A and B for the complementary angles), and label the given information.2. Find a way to relate the unknown parts to the given information using:

a. a trig function (sine, cosine, or tangent), b. the Pythagorean Theorem (a2 + b2 = c2), c. or complementary angles. d. Try to use original given information to minimize rounding errors.

3. Check your work:a. Make sure the sides obey the Pythagorean Theorem.b. Make sure the angles add up to 180.c. Make sure unused trig functions give the right answers.d. Make sure that the longest side is opposite the largest angle, and the shortest side

is opposite the smallest angle.


Section 2.5: Further Applications of Right Triangles

To solve applied right triangle problems:1. Make a sketch of the situation.2. Identify/draw right triangles on your sketch that connect given information to unknown

information.3. Solve the right triangle or triangles.

Bearing:In navigation, the word bearing means one of two things:

An angle measured clockwise from due north. An angle measured from either due north or due south in either a clockwise or

counterclockwise direction. The direction is specified as a rotation either to the east or the west. The starting direction is stated first, then the angle, then the direction of rotation of that angle from the starting direction.


Section 3.1: Radian Measure the radian measure of a central angle in a circle is the proportionality constant between

the arc length intercepted by that central angle and the radius of the circle (s = r). This implies that there are 2 radians in a full circle, because the arc length around a full

circle is the circumference: s = C = 2r.

Definition of the RadianAn angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian.

Converting Between Degrees and Radians There are 2 radians in a full circle. There are 360 in a full circle. This implies the following conversion factors:

To convert from degrees to radians, multiply by .

To convert from radians to degrees, multiply by .

If no unit of angle measure is stated, then the angle is understood to be measure in radians.


Section 3.2: Applications of Radian MeasureArc LengthThe length s of the arc intercepted on a circle of radius r by a central angle of measure radians is given by the product of the radius and the angle measure.

s = r

Area of a Sector of a CircleThe area A of a sector of a circle of radius r by a central angle of measure radians is given by:


Section 3.3: The Unit Circle and Circular Functions

Circular FunctionsStarting with a unit circle radius r = 1 and ,

So, the input to the trig functions can be thought of as either an angle or an arc length, both of which are real numbers (and not an angle measured in degrees)Also, the output of the sine and cosine functions can be interpreted as the y and x coordinates, respectively, of a point on the unit circle.

Circular Functions (Section 3.3)


Section 3.4: Linear and Angular Speed

Think about a point moving around a circle of radius r at a constant speed:

Linear speed is the rate of change of distance per unit time.

Angular speed is the rate of change of angle per unit time.

There is a relationship between angular speed and linear speed for an object going around a circle:


Section 4.1: Graphs of the Sine and Cosine FunctionsA periodic function f is a function such that

For every real number x in the domain of f, every integer n, and some positive real number p. The least possible positive value for p is called the period of the function.


Graph of the Sine Function

Domain: Range:

Notes on the graph of the sine function: The period is 2. The graph is continuous over its entire domain. The x-intercepts occur at n, where n is an integer. The sine function is an odd function .


Graph of the Cosine Function

Domain: Range:

Notes on the graph of the cosine function: The period is 2. The graph is continuous over its entire domain.

The x-intercepts occur at , where n is an integer (i.e., at odd multiples of ).

The cosine function is an even function .


Graphs of Vertically Stretched Sine and Cosine Functions

AmplitudeThe graph of or with will have the same shape as the graph of

or , except with a range of . The amplitude is .

Modified PeriodThe graph of or with will have the same shape as the graph of

or , except with a period of .

The graphs are organized around finding the locations of the zeros and extrema of the sine and cosine functions. In fact, both functions follow the pattern of:Zero Maximum Zero Minimum Zero Maximum Zero Minimum …It is just a matter of where each one starts in the cycle at x = 0…This suggests a quick algorithm for sketching these two functions:

Summary of How to Graph Stretched Sine and Cosine FunctionsTo graph or , with b > 0:

Find the period , then starting at 0 on the x-axis, mark off a couple multiples of the distance in both the positive and negative directions.

Divide each interval into four equal parts. Plot the zeros, maxima, and minima at the appropriate locations on each quarter-period. Connect the dots with a “sinusoidal-looking” curve.


Section 4.2: Translations of the Graphs of the Sine and Cosine FunctionsHorizontal TranslationsRecall from algebra that adding or subtracting a constant to the independent variable in a function shifts the graph left or right, which is called a horizontal translation.

The graph of , assuming d > 0, shifts the graph of to the right by d units.

The graph of , assuming d > 0, shifts the graph of to the left by d units.

The same result applies to the graphs of the trigonometric functions, with two new vocabulary words:

A horizontal translation is called a “phase shift”. The expression x – d that you plug into the trig function is called the “argument”.

To graph a horizontally translated trig function, you can use one of two methods: Draw the trig function shifted left or right by the given phase shift. Make a table of values where you list the key values for the argument x – d, and then

work backwards to find the x value that produces those key values.

To graph , recall that you have to factor a 2 out of both terms to get the correct phase shift.


Vertical TranslationsRecall from algebra that adding or subtracting a constant to a function shifts the graph up or down, which is called a vertical translation.

The graph of , assuming c > 0, shifts the graph of up by c units.

The graph of , assuming c > 0, shifts the graph of down by c units.

To graph a vertically translated trig function, you can use one of two methods: Draw the trig function shifted up or down. Make a table of values where you compute the value of the shifted function for key

values for the argument of the trig function.

Summary of How to Graph Linearly Transformed Sine and Cosine FunctionsTo graph or , with b > 0 using key values:

Find an interval whose length is one period by solving the compound inequality

Divide each interval into four equal parts. Plot the maxima, minima, and “middle” points (that intersect the line y = c) at the

appropriate locations on each quarter-period. Connect the dots with a “sinusoidal-looking” curve.

To graph or , with b > 0 using transformations: Identify the amplitude |a| and period . Use translations to to graph the desired functions.


Section 4.3: Graphs of the Tangent and Cotangent Functions

Graph of the Tangent Function

Domain: Range:

Notes on the graph of the tangent function: The period is .

The graph is discontinuous at x-values , where n is an integer (i.e., at odd

multiples of ). The x-intercepts occur at n, where n is an integer. The graph has no amplitude, since there are no extrema (it goes to ). The tangent function is an odd function .


Graph of the Cotangent Function

Domain: Range:

Notes on the graph of the cotangent function: The period is . The graph is discontinuous at x-values , where n is an integer.

The x-intercepts occur at n, where n is an integer.

The graph has no amplitude, since there are no extrema (it goes to ). The cotangent function is an odd function .


Section 4.4: Graphs of the Secant and Cosecant Functions

Graph of the Secant Function

Domain: Range:

Notes on the graph of the secant function: The period is 2.

The graph is discontinuous at x-values , where n is an integer (i.e., at odd

multiples of ). There are no x-intercepts. The graph has no amplitude, since it goes to . The graph has “local” extrema of 1 at integer multiples of

The secant function is an even function .


Graph of the Cosecant Function

Domain: Range:

Notes on the graph of the cosecant function: The period is 2. The graph is discontinuous at integer multiples of . There are no x-intercepts. The graph has no amplitude, since it goes to . The graph has “local” extrema of 1 at odd multiples of

The cosecant function is an odd function .


Section 4.5: Harmonic MotionSimple Harmonic Motion

Simple harmonic motion occurs when an object is moving under the influence of a force whose strength is proportional to the distance the object has moved from its rest position, and that always points toward the rest position (F = -kx).

The prototypical example is a mass bouncing up and down as it hangs on a spring.o When the mass is not moving, it stretches the spring to its rest position.o If the mass is pulled down or pushed up from the rest position by a distance a, it

will bounce up and down after it is released.

o The amplitude of the motion is |a|.o The time it takes for one full oscillation (from the top rest position bottom

rest position top) is called the period. The period is the time for one cycle, and is frequently labeled T. T is usually measured in seconds.

o The frequency is how many complete oscillations, or cycles, occur per unit time. Frequency is frequently labeled f. f is measured in cycles per second. This unit of measure is called the Hertz, in honor of the German physicist Heinrich Hertz.

o Frequency is the reciprocal of period: ;

o Period is the reciprocal of frequency:

o The angular frequency of the oscillation is frequently labeled , and is used to convert from the “regular” frequency measured in cycles per second to a measure in radians per second so that we can model the oscillation using sine or cosine as a function of time:


o This bouncing would continue forever if it weren’t for air drag, friction, etc.

Simple Harmonic Motion Equations:The position of an object oscillating about an equilibrium position as a function of time t can be modeled by either:

or Where a and are constants, with > 0. The amplitude of the motion is |a|, the period is

, and the frequency is .

Section 5.1: Fundamental IdentitiesReciprocal Identities:

Quotient Identities:

Pythagorean Identities:

Negative-Angle Identities:


Section 5.2: Verifying Trigonometric Identities

Hints for Verifying Identities Learn (i.e., memorize) the fundamental identities from section 5.1, and be aware of their

equivalent forms (like a re-arranged Pythagorean identity.Reciprocal Identities:

Quotient Identities:

Pythagorean Identities:

Negative-Angle Identities:

Try to simplify the more complicated side until it looks like the simpler side. Sometimes it is helpful to express all trig functions in terms of sine and cosine, and then

simplify. Usually it helps to factor when possible, and perform any indicated algebraic operations.

Example: replace with its factored form of

Example: replace with the fraction of

As you make substitutions to convert one side into the other, always work toward the goal of the other side.

A common trick is to multiply expressions like by the following fraction, because then a Pythagorean identity can be used to simplify:


Section 5.3: Sum and Difference Identities for Cosine

Cosine of a Sum or Difference

Cofunction Identities

Section 5.4: Sum and Difference Identities for Sine and Tangent

Sine of a Sum or Difference

Tangent of a Sum or Difference


Section 5.5: Double-Angle Identities

Double Angle Identities

Product-To-Sum Identities

Sum-To-Product Identities


Section 5.6: Half-Angle Identities

Half-Angle Identities

Section 6.1: Inverse Circular FunctionsVertical Line testAny vertical line will intersect the graph of a function in at most one point.

Horizontal Line TestAny horizontal line will intersect the graph of a one-to-one function in at most one point.

Inverse FunctionThe inverse function of a one-to-one function f is defined as . In other words, switch the x and y values of points on the graph of a function to obtain the graph of an inverse function.

Summary of Inverse Functions For a one-to-one function, each x-value corresponds to only one y-value, and each y-

value corresponds to only one x-value (i.e., the function passes the vertical line test and the horizontal line test).

If a function f is one-to-one, then f has an inverse function, which we write as f-1. The domain of f is the range of f-1, and the range of f is the domain of f-1. The graphs of f and f-1 are reflections of each other across the line y = x. To find f-1(x) from an algebraic function f(x), follow these steps:

o Interchange x and y in the equation y = f(x).o Solve for y.o The resulting expression that y is equal to is f-1(x).


Inverse Sine Functionor means that , for .

Note that the domain of the sine function has to be restricted to make it a one-to-one function in order to define an inverse.

Graph of the Inverse Sine FunctionOR

Domain: Range:

Table of Values:

x

Notes on the graph of the inverse sine function: The inverse sine function is increasing and continuous on its domain. Both the x- and y- intercepts are 0. The inverse sine function is an odd function.


Inverse Cosine Functionor means that , for .

Note that the domain of the cosine function has to be restricted to make it a one-to-one function in order to define an inverse.

Graph of the Inverse Cosine FunctionOR

Domain: Range:

Table of Values:

x

Notes on the graph of the inverse cosine function: The inverse cosine function is decreasing and continuous on its domain. Its x-intercept is 1, and its y-intercept is /2. The inverse cosine function is neither odd nor even.


Inverse Tangent Functionor means that , for .

Note that the domain of the tangent function has to be restricted to make it a one-to-one function in order to define an inverse.

Graph of the Inverse Tangent FunctionOR

Domain: Range:

Table of Values:

x

Notes on the graph of the inverse tangent function: The inverse tangent function is increasing and continuous on its domain. Both the x- and y- intercepts are 0. The inverse tangent function is odd. The lines are horizontal asymptotes.


Inverse Cotangent, Secant, and Cosecant Functions


Finding Inverse Trigonometric Functions with a Calculator

Finding Trigonometric Functions of Inverse Trigonometric Functions(Note: there are restrictions to the domains and ranges of the formulas below that are being glossed over…)

Section 6.2: Trigonometric Equations I


To solve a trigonometric equation: If possible, graph the equation first so you can see what kind of answers to expect. Use identities so that only one of the trig functions occurs in the equation, and every

occurrence of that trig function has the same argument. o Sometimes factoring first can lead to “mini equations,” each of which has only

one trig function, which means you don’t need to use trig identities. Use algebra to isolate the trig function Use inverse trig functions and the unit circle to find the angle or angles that solve the

equation. Check your answers; sometimes an extraneous solution is introduced when you square

both sides of the equation.

Section 6.3: Trigonometric Equations IITo solve a trigonometric equation:

If possible, graph the equation first so you can see what kind of answers to expect. Use identities so that only one of the trig functions occurs in the equation, and every

occurrence of that trig function has the same argument. o Sometimes factoring first can lead to “mini equations,” each of which has only

one trig function, which means you don’t need to use trig identities. Use algebra to isolate the trig function Use inverse trig functions and the unit circle to find the angle or angles that solve the

equation. Check your answers; sometimes an extraneous solution is introduced when you square



Section 6.4: Equations Involving Inverse Trigonometric FunctionsTo solve an inverse trigonometric equation:

If possible, graph the equation first so you can see what kind of answers to expect. Use algebra to isolate one of the inverse trig functions. Take the corresponding trig function of both sides of the equation.

o If necessary, use angle sum or difference formulas.o If necessary, use the trick from section 6.1 to compute an exact algebraic

expression for the composition of an inverse trig function and a trig function. Check your answers; sometimes an extraneous solution is introduced when you square


Section 7.1: Oblique Triangles and the Law of Sines

The Law of Sines

OR

Area of a Triangle

OR

OR


Section 7.2: The Ambiguous Case of the Law of Sines

Suppose you are asked to follow the following three steps to draw a triangle: Draw a 50 angle to form one vertex of the triangle.

Measure off 3.0 cm on one of the sides of the angle to form a side of the triangle.

From the new vertex formed by that side, draw a segment that is 2.5 cm long such that its endpoint intersects the other unused side of the angle.

o Some people would draw the 2.5 cm side “kicked out” to the right:

o Some people would draw the 2.5 cm side “kicked in” to the left:


A valid triangle is formed in both cases, but the directions are ambiguous as to which one is desired. The point of this section is to be able to calculate the remaining side and angles for both triangles.

The relationships between the angles can be visualized when both triangles are drawn on top of each other:

TO FIND THE OTHER ANGLE for the ambiguous case, subtract the angle from the laws of sines from 180.


Section 7.3: The Law of Cosines

The Law of Cosines

OR

OR

OR

OR


Section 7.4: Vectors, Operations, and the Dot ProductVocabulary:

The line segment pointing from the point (1,1) to the point (3,3) has a length (or magnitude) of and a direction of 45 above the horizontal.

The point (1, 1) is called the initial point of the vector, and the point (3,3) is called the terminal point of the vector.

To name this directed line segment as a vector, we pick a variable, and then write it using either bold face type, or with an arrow over the letter. So, if we want to use the letter A we would write the vector as A or .

Notice that the vector pointing from (-1, 0) to (1, 2) also has a magnitude of and a direction of 45 above the horizontal. We say that this is an equivalent vector to A because it has the same magnitude and direction; the location of the initial point just doesn’t matter.

When we only want to talk about the magnitude of a vector, we either put the vector in absolute value bars, or write its name without the bold face (or arrow). So, for vector A, we would write , or .


There are two ways to state both the measurements of a vector.o The first way is to state the magnitude and direction outright:

o The second way is to notice that you can get from the initial point to the terminal

point by moving 2 units in the x-direction followed by 2 units in the y-direction. These distances are called the components of the vector, and there are 3 common ways to write a vector using its components.

(this is the style our book uses)

To add vectors graphically, place them “tip-to-tail,” because adding vectors can be thought of as a sequence of motions.

The vector that results from drawing a vector from the first initial point to the last terminal point is called the resultant vector, and many times is named R.

Practice:1. Suppose you follow a vector A = 110 mi. at 20 and the follow a vector B = 220 mi. at

225. Your final location is A + B. State an approximate value for A + B. Also state what B + A is.


To add vectors mathematically, we add like components, because we can get from the first initial point to the last terminal point by moving in the x and y directions only.

The horizontal and vertical components of a vector u with magnitude |u| and direction angle are given by:

and

This means that

If you only have the components of a vector, then the magnitude and direction can be found from:

and

Effect of multiplying a scalar and a vector:The vector gets stretched, compressed, or reversed.

The Dot ProductThe dot product of two vectors and is defined by

.

The dot product is handy because it also has a geometric interpretation:

Geometric Interpretation of the Dot ProductIf is the angle between two vectors a and b, then .

This can be used to find the angle between to vectors.

Section 7.5: Applications of Vectors1. In this section, translate the word problem into vectors, then do vector addition.2. Physics: sum of forces = 0 for a body at rest or moving at constant velocity.3. Physics: sum of forces = mass times acceleration for accelerating bodies.


Section 8.1: Complex Numbers Recall that for any real number a, a2 0. Examples:

o

o

o Consequence: there is no real number whose square is less than zero.

o i.e., is not a real number.o Mathematicians have found it useful to define a new type of number, called the

imaginary unit, whose square is equal to negative one. This opened up a floodgate of new knowledge and advanced applications.

Definition: The imaginary UnitThe imaginary unit, denoted by i, is the number whose square is -1. That is:

i2 = -1OR

Definition: Complex NumbersA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. The real number a is called the real part of the complex number, and the real number b is called the imaginary part of the complex number.

The standard form for writing a complex number is the form a + bi. A pure imaginary number is a complex number of the form bi. Note: under this definition, the real numbers are a subset of the complex numbers.

The Expression If a is a positive real number, then we define the principal square root of –a, denoted as , as

.


THE ARITHMETIC OF COMPLEX NUMBERSNote: Adding, subtracting, multiplying, or dividing complex numbers results in an answer that is also a complex number.

To equate complex numbers, equate “like terms”:

To add complex numbers, add “like terms”:

To subtract complex numbers, subtract “like terms”:

To multiply complex numbers, use the distributive property, then combine “like terms” (like using FOIL):

To multiply complex numbers, keep in mind that the product property of radicals does not apply when the radicand is a negative number, so you need to evaluate the square root of the negative number first, then multiply the radicals.

Correct:

Incorrect:

The Complex ConjugateThe conjugate of a complex number z, which is commonly written as , is defined as the complex number with the same real part as z, but an imaginary part that is the negative of the imaginary part of z.If

, then

.


Product Property of Complex ConjugatesThe product of a complex number and its conjugate is the sum of the squares of the parts of the complex number.If

then

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator, then multiply and simplify. Note: the denominator will always multiply out to a difference of squares, which will be a real number.

Example:

To compute powers of i, simplify by writing i to a power that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, as can be seen from making a list of powers of i.

i0 = i4 = i8 =

i1 = i5 = i9 =


i2 = i6 = i10 =

i3 = i7 = i11 =

Section 8.2: Trigonometric (Polar) Form of Complex Numbers

Relationships Among x, y, r, and

Trigonometric (Polar) Form of a Complex NumberThe expression

Is called the trigonometric form (or polar form) of a complex number . The expression is sometimes abbreviated can be written .


8.3: The Product and Quotient TheoremsProduct TheoremIf and , then

.

Quotient TheoremIf and , then

.

Section 8.4: DeMoivre's Theorem; Powers and Roots of Complex NumbersDeMoivre’s TheoremIf and n is any real number, then

.nth RootFor a positive integer n, the complex number is the nth root of a complex number z if

.nth Root TheoremFor a positive integer n, the complex roots of a complex number

are given by:

trigonometry final exam review, spring 2010faculty.madisoncollege.edu/kmirus/20092010b/804213… ·...

Documents