SECTION 14-4

• Right Triangles and Function Values

Slide 14-4-1

RIGHT TRIANGLES AND FUNCTION VALUES

• Right Triangle Side Ratios• Cofunction Identities• Trigonometric Function Values of Special Angles• Reference Angles

Slide 14-4-2

RIGHT TRIANGLE SIDE RATIOS

Slide 14-4-3

The next slide shows an acute angle A in standard position. The definitions of the trigonometric function values of angle A require x, y, and r. x and y are the lengths of the two legs of the right triangle ABC, and r is the length of the hypotenuse. The functions of trigonometry can be adapted to describe the ratios of these sides.

RIGHT TRIANGLE SIDE RATIOS

Slide 14-4-4

B

y

xx

y

r (x, y)

A C

The side of length y is called the side opposite angle A, and the side of length x is called the side adjacent to angle A.

RIGHT TRIANGLE SIDE RATIOS

Slide 14-4-5

The lengths of these sides can be used to replace x and y in the definitions of the trigonometric functions, with r replaced by the length of the hypotenuse.

RIGHT TRIANGLE-BASED DEFINITIONS OF TRIGONOMETRIC FUNCTIONS

Slide 14-4-6

side opposite sinhypotenuse

y Ar

side adjacent to coshypotenuse

x Ar

side opposite tanside adjacent to

y Ax A

For any acute angle A in standard position,

more

RIGHT TRIANGLE-BASED DEFINITIONS OF TRIGONOMETRIC FUNCTIONS

Slide 14-4-7

side adjacent to cot .side opposite

x Ay A

hypotenusesecside adjacent to

rx A

hypotenusecscside opposite

ry A

EXAMPLE: FINDING TRIGONOMETRIC FUNCTION VALUES OF AN ACUTE ANGLE

Slide 14-4-8

A

B

C

513

12

Find the values of the trigonometric functions for angle A in the right triangle.

5sin13

A 12cos13

A 5tan

12A

13csc5

A 13sec12

A 12cot5

A

Solution

COFUNCTION IDENTITIES

Slide 14-4-9

sin cosaA Bc

tan cotaA Bb

sec csccA Bb

A

B

C

a

b

c

COFUNCTION IDENTITIES

Slide 14-4-10

Because C = 90°, A and B are complementary angles. Because A and B are complementary angles and sin A = cos B, the functions sine and cosine are called cofunctions. Also, tangent and cotangent are cofunctions, as are secant and cosecant. And because A and B are complementary angles, we have B = 90° – A. This leads to

sin A = cos B = cos(90° – A).

The rest of the cofunction identities are on the next slide.

COFUNCTION IDENTITIES

Slide 14-4-11

sin cos(90 )A A csc sec(90 )A A

cos sin(90 )A A

tan cot(90 )A A

sec csc(90 )A A

cot tan(90 )A A

For any acute angle A,

EXAMPLE: WRITING FUNCTIONS IN TERMS OF COFUNCTIONS

Slide 14-4-12

Write each of the following in terms of cofunctions.

a) cos 48° b) tan 33° c) sec 81°

a) sin 42°

b) cot 57°

c) csc 9°

Solution

TRIGONOMETRIC FUNCTION VALUES OF SPECIAL ANGLES

Slide 14-4-13

Certain special angles, such as 30°, 45°, and 60°, occur so often in applications of trigonometry that they deserve special study. The exact trigonometric function values of these angles, found by the properties of geometry and the Pythagorean theorem, are summarized on the next slide.

TRIGONOMETRIC FUNCTION VALUES OF SPECIAL ANGLES

Slide 14-4-14

30° 2

45° 1 1

60° 2

sin cos tan cot sec csc

12

12

3

3

2 222

22

32

32

33

33

2 33

2 33

REFERENCE ANGLES

Slide 14-4-15

Associated with every nonquadrantal angle in standard position is a positive acute angle called its reference angle. A reference angle for an angle written is the positive acute angle made by the terminal side of angle and the x-axis.

,,

x

x

yy

EXAMPLE: REFERENCE ANGLES

Slide 14-4-16

Find the reference angle for 232°

Solution

x

232 180 52

y

232°

52°

EXAMPLE: REFERENCE ANGLES

Slide 14-4-17

Find the reference angle for 1020°

Solution

360 300 60

x

y

300°

60°

Find a coterminal angle between 0° and 360°: 1020° – 2(360°) = 300°

REFERENCE ANGLES, WHERE

Slide 14-4-18

0 360

x

x

y

y

x

y

x

180 y

Q III

Q II 180

360

Q I

Q IV

FINDING TRIGONOMETRIC FUNCTION VALUES FOR ANY NONQUADRANTAL ANGLE

Slide 14-4-19

Step 1 If > 360°, or if < 0°, find a coterminal angle by adding or subtracting 360° as many times as needed to obtain an angle greater than 0° but less than 360°.

Step 2 Find the reference angle

Step 3 Find the necessary values of the trigonometric functions for the reference angle

.

.

FINDING TRIGONOMETRIC FUNCTION VALUES FOR ANY NONQUADRANTAL ANGLE

Slide 14-4-20

Step 4 Determine the correct signs for the values found in step 3. This result gives the values of the trigonometric functions for angle .

EXAMPLE: FINDING TRIGONOMETRIC FUNCTION VALUES USING A REFERENCE ANGLE

Slide 14-4-21

Use a reference angle to find the exact value of cos 495°.

Solution

180 135 45

Find a coterminal angle between 0° and 360°: 495° – 360° = 135°

2cos 495 cos135 cos 452

In quadrant II, so cosine is negative.