Trigonometry

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Chapter 2Chapter 2

TrigonometricTrigonometricFunctionsFunctions

Trigonometry

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Chapter 2Chapter 2OverviewOverview

Trigonometry

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Chapter 2Chapter 2ObjectivesObjectives

Draw angles in the Cartesian plane. Define trigonometric functions as ratios of x and

y coordinates and distances in the Cartesian plane.

Evaluate trigonometric functions for nonacute angles.

Determine ranges for trigonometric functions and signs for trigonometric functions in each quadrant.

Derive and use basic trigonometric identities.

Trigonometry

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Section 2.1Section 2.1Angles in the Cartesian PlaneAngles in the Cartesian Plane

Skills Objectives Plot angles in

standard position. Identify coterminal

angles. Graph common

angles.

Conceptual Objectives Relate the x and y

coordinates to the legs of a right triangle.

Derive the distance formula from the Pythagorean Theorem.

Connect angles with quadrants.

Trigonometry

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Angles in Standard PositionAngles in Standard Position

An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at the origin.

We say that an angle lies in the quadrant in which its terminal side lies.

Trigonometry

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Sketching Angles in Standard PositionsSketching Angles in Standard Positions

Sketching a 210º angle in the standard position yields this graph.

•The initial side lies on the x-axis.•The positive angle indicates counterclockwise rotation.•180º represents a straight angle and the additional 30º yields a 210 º angle.•The terminal side lies in quadrant III.

Trigonometry

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Coterminal anglesCoterminal angles

Two angles in standard position with the same terminal side are called coterminal angles. For example, -40º and 320º are coterminal angles. Moving 40º in clockwise direction brings the terminal side to the same position as moving 320º in the counter-clockwise direction.

Such angles may also be reached by going the same direction, such as 90º and 450º. 450º is reached by moving counterclockwise through the full 360º circle, then continuing another 90 º.

Trigonometry

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Coterminal AnglesCoterminal Angles

If you graph angles x = 30o and y = - 330o in standard position, these angles will have the same terminal side. See figure below

Coterminal angles Ac to angle A may be obtained by adding or subtracting k*360 degrees or k* (2π).

Hence Ac = A + k*360o if A is given in degrees. Or Ac = A + k*(2π) if A is given in radians; where k is any negative or positive integer.

Trigonometry

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Your Turn:Your Turn:Measuring of Coterminal AnglesMeasuring of Coterminal Angles

580º Solution: Subtract 360º to find the correct

angle of 220º. -400º Solution: Add 360º to get -40º. Add 360º

again to get the correct angle of 320º.

Determine the smallest possible measure of these angles:

Trigonometry

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Common Angles in Standard PositionCommon Angles in Standard Position

Standard Position:  An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis.  The ray on the x-axis is called the initial side and the other ray is called the terminal side.  If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle.  The angle shown at the right is referred to as a Quadrant II angle since its terminal side lies in Quadrant II.

If Θ is an angle in standard position, and P is any point (other than the origin) on the terminal side of Θ, then we associate 3 numbers with the point P.

x: x-coordinate of the point P y: y-coordinate of the point P r : distance of the point from the origin

Trigonometry

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The common angles with their exact values

for their Cartesian coordinates are shown on this graph.

Common Angles in Standard PositionCommon Angles in Standard Position

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Section 2.2 Section 2.2 Definition 2 of Trigonometric Functions: Definition 2 of Trigonometric Functions:

Cartesian PlaneCartesian Plane

Skills Objectives Calculate trigonometric function

values for acute angles. Calculate trigonometric function

values for nonacute angles. Calculate trigonometric function

values for quadrantal angles.

Conceptual Objectives Define trigonometric functions in

the Cartesian plane. Extend right triangle definitions

of trigonometric functions for acute angles to definitions of trigonometric functions for all angles in the Cartesian plane.

Understand why some trigonometric functions are undefined for quadrantal angles.

Trigonometry

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The Cartesian PlaneThe Cartesian Plane

Line up a right triangle with a perpendicular segment connecting the point (x, y) to the x-axis.

The distance from the origin, (0, 0), to the point (x, y) is now:

2222 )0()0( yxyxr

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Trigonometric FunctionsTrigonometric Functions

All of the trigonometric functions are defined by the values of the three sides of a right triangle.

Trigonometry

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Calculating Trigonometric Function ValuesCalculating Trigonometric Function Values

For this angle x = 2 and y = 5.

The distance from the origin is

.

sinθ = =

cosθ = =

tanθ = =

The remainder are calculated from

these three values.

29

r

y29

5

29

2

2

5r

x

x

y

Trigonometry

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Your Turn : Calculating Trigonometric Your Turn : Calculating Trigonometric Functions for Nonacute AnglesFunctions for Nonacute Angles

Calculate the values of x, y,

and r in the same way.

r must be positive.

For this graph x = -1, y = -3,

and r = .

Click for answers!

10

Trigonometry

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Your Turn : Calculating Trigonometric Your Turn : Calculating Trigonometric Functions for Nonacute AnglesFunctions for Nonacute Angles

Calculate the values of x, y, and r in

the same way.

r must be positive.

For this graph x = -1, y = -3, and

r = .

sinθ = =

cosθ = =

tanθ = = 3

10

r

y

r

x

x

y

10

3

10

1

Trigonometry

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Quadrantal ValuesQuadrantal Values

Θ SINΘ COSΘ TANΘ COTΘ SECΘ CSCΘ

0° 0 1 0 U 1 U

90° 1 0 U 0 U 1

180° 0 -1 0 U -1 U

270° -1 0 U 0 U -1

360° 0 1 0 U 1 U

The table below summarizes the trigonometric function values for common quadrantal angles: 0°, 90 °, 180 °, 270 °, and 360 °.

Trigonometry

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Section 2.3 Section 2.3 Trigonometric Functions of Nonacute AnglesTrigonometric Functions of Nonacute Angles

Skills Objectives Determine the reference

angle of a nonacute angle.

Evaluate trigonometric functions exactly for common angles.

Approximate trigonometric functions of nonacute angles.

Conceptual Objectives Determine algebraic signs of

trigonometric functions for all four quadrants.

Determine values for trigonometric functions for quadrantal angles.

Determine ranges for trigonometric functions.

Trigonometry

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Algebraic Signs of Trigonometric Algebraic Signs of Trigonometric FunctionsFunctions

POSITIVEAllStudentsTakeCalculus

Sin Θ = y/r

Cos Θ = x/r

Tan Θ = y/x

Csc Θ = r/y , y≠ 0

Sec Θ = r/x , x ≠ 0

Cot Θ = x/y , y ≠ 0

°y

r

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Using the Algebraic Sign of a Trigonometric Using the Algebraic Sign of a Trigonometric FunctionFunction

If cosθ = -3/5 and the terminal side of the angle lies in quadrant III, find sinθ.cosθ = -3/5 means that the x value is negative, so x = -3 and r = 5.Now we know that (-3)2 + y2 = 52.y2 = 25 – 9 = 16, so y = ±4.Since the angle is in quadrant III, y = -4.sinθ = y/r = -4/5.

Trigonometry

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Values of Quadrantal Trigonometric Values of Quadrantal Trigonometric FunctionsFunctions

The values of the trigonometric functions for angles along the axes are undefined for some angles. For example, along the positive y-axis, the value of x is zero, making the value of the tangent undefined.

Trigonometry

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Reference TriangleReference Triangle

A reference triangle is formed by "dropping" a perpendicular from the terminal ray of a standard position angle to the x-axis.   Remember, it must be drawn to the x-axis.Reference triangles are used to find trigonometric values for their standard position angles.  They are of particular importance for standard position angles whose terminal sides reside in quadrants II, III and IV.  A reference triangle contains a reference angle.

Trigonometry

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Section 2.4 Section 2.4 Basic Trigonometric IdentitiesBasic Trigonometric Identities

Skills Objectives Learn the reciprocal

identities. Learn the quotient

identities. Learn the Pythagorean

identities. Use the basic identities to

simplify expressions.

Conceptual Objectives Understand that

trigonometric reciprocal identities are not always defined.

Understand that quotient identities are not always defined.

Trigonometry

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Reciprocal IdentitiesReciprocal Identities

Since sinθ = y/r and cscθ = r/y, these two trigonometric functions are reciprocals of one another. Therefore, if y ≠ 0, then cscθ is defined.

Similarly, cosθ = x/r and secθ = r/x(defined if x ≠ 0) are reciprocal functions as are

tanθ = y/x (defined if x ≠ 0) and cotθ = x/y (defined if y ≠ 0) .

Trigonometry

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Quotient IndentitiesQuotient Indentities

Since tanθ = sinθ /cosθ and

cotθ = cosθ /sinθ, these two trigonometric functions are called quotient identities.

Therefore, if cosθ ≠ 0, then tanθ is defined

and if sinθ ≠ 0, then cotθ is defined.

Trigonometry

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Pythagorean IdentitiesPythagorean Identities

Pythagorean Identities Variations

Sin Θ 2 + Cos Θ 2 = 1 Sin Θ 2 = 1 - Cos Θ 2 Cos Θ 2 = 1 - Sin Θ 2

Tan Θ 2 + 1 = Sec Θ 2 Tan Θ 2 = Sec Θ 2 - 1

1 + Cot Θ 2 = Csc Θ 2 Cot Θ 2 = Csc Θ 2 - 1

When studying the unit circle, it was observed that a point on the unit circle (the vertex of the right triangle)

can be represented by the coordinates (cos Θ, sin Θ ). 

Since the legs of the right triangle in the unit circle have the values of cos Θ and sin Θ, the Pythagorean Theorem can be used to obtain …. .