Chapter 5 Section 3

5.3-1

TrigonometricFunctions

5.3-2

5.1 Angles

5.2 Trigonometric Functions

5.3 Evaluating Trigonometric Functions

5.4Solving Right Triangles

Trigonometric Functions

5.3-3

Evaluating TrigonometricFunctions

5.3

Right-Triangle-Based Definitions of the Trigonometric Functions ▪Cofunctions ▪ Trigonometric Function Values of Special Angles ▪Reference Angles ▪ Special Angles as Reference Angles ▪Finding Function Values Using a Calculator ▪ Finding AngleMeasures with Special Angles

5.3-4

Right-Triangle-Based Definitionsof Trigonometric Functions

For any acute angle A in standard position,

5.3-5

Right-Triangle-Based Definitionsof Trigonometric Functions

For any acute angle A in standard position,

5.3-6

Right-Triangle-Based Definitionsof Trigonometric Functions

For any acute angle A in standard position,

5.3-7

Find the sine, cosine,and tangent values forangles A and B.

Example 1

FINDING TRIGONOMETRIC FUNCTIONVALUES OF AN ACUTE ANGLE

5.3-8

Find the sine, cosine,and tangent values forangles A and B.

Example 1

FINDING TRIGONOMETRIC FUNCTIONVALUES OF AN ACUTE ANGLE (cont.)

5.3-9

Cofunction Identities

For any acute angle A in standard position,

sin A = cos(90  A)csc A = sec(90  A)

tan A = cot(90  A)cos A = sin(90  A)

sec A = csc(90  A)cot A = tan(90  A)

5.3-10

Write each function in terms of its cofunction.

Example 2

WRITING FUNCTIONS IN TERMS OFCOFUNCTIONS

(a)cos 52°

= sin (90° – 52°) = sin 38°

(b)tan 71°

= cot (90° – 71°) = cot 19°

(c)sec 24°

= csc (90° – 24°) = csc 66°

5.3-11

30°- 60°- 90° Triangles

Bisect one angle of an equilateralto create two 30°-60°-90° triangles.

5.3-12

30°- 60°- 90° Triangles

Use the Pythagorean theorem to solve for x.

What would x have been if the shorter side had been 5?

5.3-13

Example

FINDING TRIGONOMETRIC FUNCTIONVALUES FOR 60°

Find the six trigonometric function values for a 60°angle.

5.3-14

Example

FINDING TRIGONOMETRIC FUNCTIONVALUES FOR 60° (continued)

Find the six trigonometric function values for a 60°angle.

5.3-15

45°- 45° Right Triangles

Use the Pythagorean theorem tosolve for r.

5.3-16

45°- 45° Right Triangles

Do the trig functions depend on the length of the side?

5.3-17

45°- 45° Right Triangles

5.3-18

Function Values of SpecialAngles

60

45

30

csc 

sec 

cot 

tan 

cos 

sin 



5.3-19

Reference Angles

A reference angle for an angle θ is the positiveacute angle made by the terminal side of angle θand the x-axis.

5.3-20

Caution

A common error is to find thereference angle by using the terminalside of θ and the y-axis.

The reference angle is always foundwith reference to the x-axis.

5.3-21

Find the reference angle for an angle of 218°.

Example 3(a)

FINDING REFERENCE ANGLES

The positive acute angle madeby the terminal side of theangle and the x-axis is218° – 180° = 38°.

For θ = 218°, the reference angle θ′ = 38°.

5.3-22

Find the reference angle for an angle of 1387°.

Example 3(b)

FINDING REFERENCE ANGLES

First find a coterminal anglebetween 0° and 360°.

Divide 1387 by 360 to get aquotient of about 3.9. Begin bysubtracting 360° three times.1387° – 3(360°) = 307°.

The reference angle for 307° (and thus for 1387°) is360° – 307° = 53°.

5.3-23

5.3-24

Find the values of the six trigonometric functionsfor 210°.

Example 4

FINDING TRIGONOMETRIC FUNCTIONVALUES OF A QUADRANT III ANGLE

The reference angle for a210° angle is210° – 180° = 30°.

Choose point P on theterminal side of the angle sothe distance from the originto P is 2.

5.3-25

Example 4

FINDING TRIGONOMETRIC FUNCTIONVALUES OF A QUADRANT III ANGLE(continued)

5.3-26

Find the exact value of cos (–240°).

Example 5(a)

FINDING TRIGONOMETRIC FUNCTIONVALUES USING REFERENCE ANGLES

Since an angle of –240° is coterminal with anangle of –240° + 360° = 120°, the referenceangle is 180° – 120° = 60°.

5.3-27

Find the exact value of tan 675°.

Example 5(b)

FINDING TRIGONOMETRIC FUNCTIONVALUES USING REFERENCE ANGLES

Subtract 360° to find a coterminal anglebetween 0° and 360°: 675° – 360° = 315°.

5.3-28

Caution

When evaluating trigonometricfunctions of angles given in degrees,remember that the calculator mustbe set in degree mode.

5.3-29

Example 6

FINDING FUNCTION VALUES WITH ACALCULATOR

(a) sin 53°

Approximate the value of each expression.

(b) sec 97.977°

sec 97.977° ≈ –7.20587921

Calculators do not have a secantkey, so first find cos 97.977° andthen take the reciprocal.

5.3-30

Example 6

FINDING FUNCTION VALUES WITH ACALCULATOR (continued)

(c)

Approximate the value of each expression.

Use the reciprocal identity

5.3-31

Use a calculator to find an angle θ in the interval[0°, 90°] that satisfies each condition.

Example 7

USING INVERSE TRIGONOMETRICFUNCTIONS TO FIND ANGLES

a)cos θ = .5 We know that cos ____ = .5

To find this on the calculator we use θ =

b)Find θ if cos θ = .9211854056

c)Find θ if sec θ = 1.2228

5.3-32

Find all values of θ, if θ is in the interval [0°, 360°)and

Example 8

FINDING ANGLE MEASURES GIVEN ANINTERVAL AND A FUNCTION VALUE

Since cos θ is negative, θ must lie in quadrant II or III.

The angle in quadrant II is 180° – 45° = 135°.

The angle in quadrant III is 180° + 45° = 225°.

The absolute value of cos θ is so the referenceangle is 45°.

Example 9

Problem 141, page 529

When highway curves are designed, the outside of thecurve is often slightly elevated or inclined above theinside of the curve. This inclination is calledsuperelevation . For safety reasons, it is importantthat both the curve’s radius and superelevation arecorrect for a given speed limit. If an automobile istraveling at velocity V (in feet per second), the saferadius R for a curve with superelevation θ is modeledby the formula where f and g areconstants

Example 9

a)A roadway is being designed for automobiles traveling at 45mph.

If θ = 3°, g = 32.2, and f = 0.14, calculate R to the nearest foot.

(Hint: You will need to know that 1 mile = 5280 feet)

b) A highway curve has radius R = 1150 ft and a superelevation of

θ = 2.1°. What should the speed limit (in miles per hour) be for

this curve? Use the same values for f and g from part (a).