chapter 4 trigonometric functions. 4.1 angles & radian measure objectives –recognize & use...

CHAPTER 4

Trigonometric Functions

4.1 Angles & Radian Measure• Objectives

– Recognize & use the vocabulary of angles– Use degree measure– Use radian measure– Convert between degrees & radians– Draw angles in standard position– Find coterminal angles– Find the length of a circular arc– Use linear & angular speed to describe motion on a

circular path

Angles

• An angle is formed when two rays have a common endpt.

• Standard position: one ray lies along the x-axis extending toward the right

• Positive angles measure counterclockwise from the x-axis

• Negative angles measure clockwise from the x-axis

Angle Measure

• Degrees: full circle = 360 degrees– Half-circle = 180 degrees– Right angle = 90 degrees

Radians: one radian is the measure of the central angle that intercepts an arc equal in length to the length of the radius (we can construct an angle of measure = 1 radian!)

Full circle = 2 radians

Half circle = radians

Right angle = radians

2

Radian Measure• The measure of the angle in radians is the ratio of the

arc length to the radius

• Recall half circle = 180 degrees= radians• This provides a conversion factor. If they are equal,

their ratio=1, so we can convert from radians to degrees (or vice versa) by multiplying by this “well-chosen one.”

• Example: convert 270 degrees to radians

r

s

2

3

180270

Convert 145 degrees to radians.

4

3)4

4)3

2)2

)1

Coterminal angles

• Angles that have rays at the same spot.

• Angle may be positive or negative (move counterclockwise or clockwise) (i.e. 70 degree angle coterminal to -290 degree angle)

• Angle may go around the circle more than once (i.e. 30 degree angle coterminal to 390 degree angle)

Arc length

• Since radians are defined as the central angle created when the arc length = radius length for any given circle, it makes sense to consider arc length when angle is measured in radians

• Recall theta (in radians) is the ratio of arc length to radius

• Arc length = radius x theta (in radians)

rs

Linear speed & Angular speed

• Speed a particle moves along an arc of the circle (v) is the linear speed (distance, s, per unit time, t)

• Speed which the angle is changing as a particle moves along an arc of the circle is the angular speed.(angle measure in radians, per unit time, t)

t

sv

t

Relationship between linear speed & angular speed

• Linear speed is the product of radius and angular speed.

• Example: The minute hand of a clock is 6 inches long. How fast is the tip of the hand moving?

• We know angular speed = 2 pi per 60 minutes

rv

min6.

min5min10

2

min60

26

ininininv

4.2 Trigonometric Functions: The Unit Circle• Objectives

– Use a unit circle to define trigonometric functions of real numbers

– Recognize the domain & range of sine & cosine

– Find exact values of the trig. functions at pi/4

– Use even & odd trigonometric functions

– Recognize & use fundamental identities

– Use periodic properties

– Evaluate trig. functions with a calculator

What is the unit circle?• A circle with radius = 1 unit• Why are we interested in this circle? It provides

convenient (x,y) values as we work our way around the circle.

• (1,0), theta = 0• (0,1), theta = pi/2• (-1,0), theta = pi• (0,-1), theta = 3 pi/2• ALSO, any (x,y) point on the circle would be at the

end of the hypotenuse of a right triangle that extends from the origin, such that 122 yx

sin t and cos t

• For any point (x,y) found on the unit circle, x=cos t and y=sin t

• t = any real number, corresponding to the arc length of the unit circle

• Example: at the point (1,0), the cos t = 1 and sin t = 0. What is t? t is the arc length at that point AND since it’s a unit circle, we know the arc length = central angle, in radians. THUS, cos (0) = 1 and sin (0)=0

Relating all trigonometric functions to sin t and cos t

y

x

t

tt

xtt

ytt

x

y

t

tt

)sin(

)cos()cot(

1

)cos(

1)sec(

1

)sin(

1)csc(

)cos(

)sin()tan(

Pythagorean Identities

• Every point (x,y) on the unit circle corresponds to a real number, t, that represents the arc length at that point

• Since and x = cos(t) and y=sin(t), then

• If each term is divided by , the result is

• If each term is divided by , the result is

122 yx1sincos 22 tt

t2cos

tttt

t 2222

2

sectan1,cos

1

cos

sin1

t2sin

tttt

t 2222

2

csc1cot,sin

11

sin

cos

Given csc t = 13/12, find the values of the other 6 trig. functions of t

• sin t = 12/13 (reciprocal)

• cos t = 5/13 (Pythagorean)

• sec t = 13/5 (reciprocal)

• tan t = 12/5 (sin(t)/cos(t))

• cot t = 5/12 (reciprocal)

Trig. functions are periodic• sin(t) and cos(t) are the (x,y) coordinates

around the unit circle and the values repeat every time a full circle is completed

• Thus the period of both sin(t) and cos(t) = 2 pi

• sin(t)=sin(2pi + t) cos(t)=cos(2pi + t)

• Since tan(t) = sin(t)/cos(t), we find the values repeat (become periodic) after pi, thus tan(t)=tan(pi + t)

4.3 Right Triangle Trigonometry

• Objectives

– Use right triangles to evaluate trig. Functions

– Find function values for 30 degrees, 45 degrees & 60 degrees

– Use equal cofunctions of complements

– Use right triangle trig. to solve applied problems

3,4,6

Within a unit circle, and right triangle can be sketched

• The point on the circle is (x,y) and the hypotenuse = 1. Therefore, the x-value is the horizontal leg and the y-value is the vertical leg of the right triangle formed.

• cos(t)=x which equals x/1, therefore the cos (t)=horizontal leg/hypotenuse = adjacent leg/hypotense

• sin(t)=y which equals y/1, therefore the sin(t) = vertical leg/hypotenuse = opposite leg/hypotenuse

The relationships holds true for ALL right triangles (other 3 trig.

functions are found as reciprocals)

adjacent

opposite

hypotenuse

adjacent

hypotenuse

opposite

cos

sintan

cos

sin

Find the value of 6 trig. functions of the angles in a right triangle.

• Given 2 sides, the value of the 3rd side can be found, using Pythagorean theorem

• After side lengths of all 3 sides is known, find sin as opposite/hypotenuse

• cos = adjacent/hypotenuse

• tan = opposite/adjacent

• csc = 1/sin

• sec = 1/cos

• cot= 1/tan

Given a right triangle with hypotenuse =5 and side adjacent

angle B of length=2, find tan B

5

2)4

21

2)3

2

21)2

21)1

Special Triangles

• 30-60 right triangle, ratio of sides of the triangle is 1:2: , 2 (longest) is the length of the hypotenuse, the shortest side (opposite the 30 degree angle) is 1 and the remaining side (opposite the 60 degree angle) is

• 45-45 right triangle: The 2 legs are the same length since the angles opposite them are equal, thus 1:1. Using pythagorean theorem, the remaining side, the hypotenuse, is 2

3

3

Cofunction Identities

• Cofunctions are those that are the reciprocal functions (cofunction of tan is cot, cofunction of sin is cos, cofunction of sec is csc)

• For an acute angle, A, of a right triangle, the side opposite A would be the side adjacent to the other acute angle, B

• Therefore sin A = cos B• Since A & B are the acute angles of a right

triangle, their sum = 90 degrees, thus B=• function(A)=cofunction )90( A

A90

4.4 Trigonometric Functions of Any Angle

• Objectives

– Use the definitions of trigonometric functions of any angle

– Use the signs of the trigonometric functions

– Find reference angles

– Use reference angles to evaluate trigonometric functions

Trigonometric functions of Any Angle

• Previously, we looked at the 6 trig. functions of angles in a right triangle. These angles are all acute. What about negative angles? What about obtuse angles?

• These angles exist, particularly as we consider moving around a circle

• At any point on the circle, we can drop a vertical line to the x-axis and create a triangle. Horizontal side = x, vertical side=y, hypotenuse=r.

Trigonometric Functions of Any Angle (continued)

• If, for example, you have an angle whose terminal side is in the 3rd quadrant, then the x & y values are both negative. The radius, r, is always a positive value.

• Given a point (-3,-4), find the 6 trig. functions associated with the angle formed by the ray containing this point.

• x=-3, y=-4, r =

• (continued next slide)

525)4()3( 22

Example continued

• sin A = -4/5, cos A = -3/5, tan A = 4/3

• csc A = -5/4, sec A = -5/3, cot A = ¾

• Notice that the same values of the trig. functions for angle A would be true for the angles 360+A, A-360 (negative values)

Examining the 4 quadrants

• Quadrant I: x & y are positive– all 6 trig. functions are positive

• Quadrant II: x negative, y positive– positive: sin, csc negative: cos, sec, tan, cot

• Quadrant III: x negative, y negative– positive: tan, cot negative: sin, csc, cos, sec

• Quadrant IV: x positive, y negative– positive: cos, sec negative: sin, csc, cot, tan

Reference angles

• Angles in all quadrants can be related to a “reference” angle in the 1st quadrant

• If angle A is in quadrant II, it’s related angle in quad I is 180-A. The numerical values of the 6 trig. functions will be the same, except the x (cos, sec, tan, cot) will all be negative

• If angle A is in quad III, it’s related angle in quad I is 180+A. Now x & y are both neg, so sin, csc, cos, sec are all negative.

Reference angles cont.

• If angle A is in quad IV, the reference angle is 360-A. The y value is negative, so the sin, csc, tan & cot are all negative.

Special angles

• We often work with the “special angles” of the “special triangles.” It’s good to remember them both in radians & degrees

• If you know the trig. functions of the special angles in quad I, you know them in every quadrant, by determining whether the x or y is positive or negative

290,

445,

360,

630

4.5 Graphs of Sine & Cosine

• Objectives– Understand the graph of y = sin x– Graph variations of y = sin x– Understand the graph of y = cos x– Graph variations of y = cos x– Use vertical shifts of sin & cosine curves– Model periodic behavior

Graphing y = sin x

• If we take all the values of sin x from the unit circle and plot them on a coordinate axis with x = angles and y = sin x, the graph is a curve

• Range: [-1,1]• Domain: (all reals)

Graphing y = cos x

• Unwrap the unit circle, and plot all x values from the circle (the cos values) and plot on the coordinate axes, x = angle measures (in radians) and y = cos x

• Range: [-1,1]• Domain: (all reals)

Comparisons between y=cos x and y=sin x

• Range & Domain: SAME– range: [-1,1], domain: (all reals)

• Period: SAME (2 pi)• Intercepts: Different

– sin x : crosses through origin and intercepts the x-axis at all multiples of

– cos x: intercepts y-axis at (0,1) and intercepts x-axis at all odd multiples of

,...)3,2,,0,,2,3(....,

,...

2

3,2,2

,2

3...,

2

Amplitude & Period• The amplitude of sin x & cos

x is 1. The greatest distance the curves rise & fall from the axis is 1.

• The period of both functions is 2 pi. This is the distance around the unit circle.

• Can we change amplitude? Yes, if the function value (y) is multiplied by a constant, that is the NEW amplitude, example: y = 3 sin x

Amplitude & Period (cont)

• Can we change the period? Yes, the length of the period is a function of the x-value.

• Example: y = sin(3x)– The amplitude is still 1.

(Range: [-1,1])– Period is

3

2

Phase Shift

• The graph of y=sin x is “shifted” left or right of the original graph

• Change is made to the x-values, so it’s addition/subtraction to x-values.

• Example: y = sin(x- ), the graph of y=sin x is shifted right

3

3

Vertical Shift

• The graph y=sin x can be shifted up or down on the coordinate axis by adding to the y-value.

• Example: • y = sin x + 3 moves

the graph of sin x up 3 units.

Graph y = 2cos(x- ) - 2

• Amplitude = 2• Phase shift = right• Vertical shift = down 2

4

4

4.6 Graphs of Other Trigonometric Functions

• Objectives– Understand the graph of y = tan x– Graph variations of y = tan x– Understand the graph of y = cot x– Graph variations of y = cot x– Understand the graphs of y = csc x and y = sec x

y = tan x

• Going around the unit circle, where the y value is 0, (sin x = 0), the tangent is undefined.

• At x = the graph of y = tan x has vertical asymptotes

• x-intercepts where cos x = 0, x =

,...)2

3,2,

2,

2

3(...

,...)2,,0,,2(...

Characteristics of y = tan x

• Period = • Domain: (all reals except odd multiples of • Range: (all reals)• Vertical asymptotes: odd multiples of • x – intercepts: all multiples of • Odd function (symmetric through the origin, quad

I mirrors to quad III)

2

Transformations of y = tan x

• Shifts (vertical & phase) are done as the shifts to y = sin x

• Period change (same as to y=sin x, except the original period of tan x is pi, not 2 pi)

Graph y = -3 tan (2x) + 1

• Period is now pi/2

• Vertical shift is up 1

• -3 impacts the “amplitude”

• Since tan x has no amplitude, we consider the point ½ way between intercept & asymptote, where the y-value=1. Now the y-value at that point is -3.

• See graph next slide.

Graph y = -3 tan (2x) + 1

Graphing y = cot x

• Vertical asymptotes are where sin x = 0, (multiples of pi)

• x-intercepts are where cos x = 0 (odd multiples of pi/2)

y = csc x

• Reciprocal of y = sin x• Vertical tangents where sin x = 0 (x = integer

multiples of pi)• Range: • Domain: all reals except integer multiples of pi• Graph on next slide

),1[]1,(

Graph of y = csc x

y = sec x

• Reciprocal of y = cos x• Vertical tangents where cos x = 0 (odd multiples

of pi/2)• Range: • Domain: all reals except odd multiples of pi/2• Graph next page

),1[]1,(

Graph of y = sec x

4.7 Inverse Trigonometric Functions

• Objectives– Understand the use the inverse sine function– Understand and use the inverse cosine function– Understand and use the inverse tangent function– Use a calculator to evaluate inverse trig. functions– Find exact values of composite functions with

inverse trigonometric functions

What is the inverse sin of x?

• It is the ANGLE (or real #) that has a sin value of x.• Example: the inverse sin of ½ is pi/6 (arcsin ½ = pi/6)• Why? Because the sin(pi/6)= ½• Shorthand notation for inverse sin of x is arcsin x or

• Recall that there are MANY angles that would have a sin value of ½. We want to be consistent and specific about WHICH angle we’re referring to, so we limit the range to (quad I & IV)

x1sin

2,

2

Find the domain of y =

• The domain of any function becomes the range of its inverse, and the range of a function becomes the domain of its inverse.

• Range of y = sin x is [-1,1], therefore the domain of the inverse sin (arcsin x) function is [-1,1]

x1sin

Trigonometric values for special angles

• If you know sin(pi/2) = 1, you know the inverse sin(1) = pi/2

• KNOW TRIG VALUES FOR ALL SPECIAL ANGLES (once you do, you know the inverse trigs as well!)

Find

4)4

4

3)3

4

7)2

4)1

2

2sin 1

Graph y = arcsin (x)

The inverse cosine function

• The inverse cosine of x refers to the angle (or number) that has a cosine of x

• Inverse cosine of x is represented as arccos(x) or

• Example: arccos(1/2) = pi/3 because the cos(pi/3) = ½

• Domain: [-1,1] • Range: [0,pi] (quadrants I & II)

x1cos

Graph y = arccos (x)

The inverse tangent function

• The inverse tangent of x refers to the angle (or number) that has a tangent of x

• Inverse tangent of x is represented as arctan(x) or

• Example: arctan(1) = pi/4 because the tan(pi/4)=1

• Domain: (all reals) • Range: [-pi/2,pi/2] (quadrants I & IV)

x1tan

Graph y = arctan(x)

Evaluating compositions of functions & their inverses

• Recall: The composition of a function and its inverse = x. (what the function does, its inverse undoes)

• This is true for trig. functions & their inverses, as well ( PROVIDED x is in the range of the inverse trig. function)

• Example: arcsin(sin pi/6) = pi/6, BUT arcsin(sin 5pi/6) = pi/6

• WHY? 5pi/6 is NOT in the range of arcsin x, but the angle that has the same sin in the appropriate range is pi/6

4.8 Applications of Trigonometric Functions

• Objectives

– Solve a right triangle.

– Solve problems involving bearings.

– Model simple harmonic motion.

Solving a Right Triangle• This means find the values of all angles and all side

lengths.• Sum of angles = 180 degrees, and if one is a right

angle, the sum of the remaining angles is 90 degrees.

• All sides are related by the Pythagorean Theorem:

• Using ratio definition of trig functions (sin x = opposite/hypotenuse, tan x = opposite/adjacent, cos x = adjacent/hypotenuse), one can find remaining sides if only one side is given

222 cba

Example: A right triangle has an hypotenuse = 6 cm with an angle =

35 degrees. Solve the triangle.• cos(35 degrees) = .819 (using calculator)• cos(35 degrees) = adjacent/6 cm• Thus, .819 = adjacent/6 cm, adjacent = 4.9 cm• Remaining angle = 55 degrees• Remaining side:

cma

a

a

12

122436

6)9.4(2

222

Trigonometry & Bearings

• Bearings are used to describe position in navigation and surveying. Positions are described relative to a NORTH or SOUTH axis (y-axis). (Different than measuring from the standard position, the positive x-axis.)

• means the direction is 55 degrees from the north toward the east (in quadrant I)

• means the direction is 35 degrees from the south toward the west (in quadrant III)

EN 55

WS 35