9.3 Trigonometric Functions of Any Angle

9.3

Trigonometric Functions of Any Angle

Essential Question How can you use the unit circle to define the

trigonometric functions of any angle?

Let be an angle in standard position with (x, y) a point on the terminal side of and

r = -- x2 + y2 0. The six trigonometric functions of are defined as shown.

sin = --yr

csc = --yr, y 0

y

(x, y)

cos = --xr tan = --xy, x 0

sec = --xr, x 0 cot = --xy, y 0

r

x

Writing Trigonometric Functions

Work with a partner. Find the sine, cosine, and tangent of the angle in standard position whose terminal side intersects the unit circle at the point (x, y) shown.

( a. -1 , 3 2 2

y

( b. -1 , 1 22

y

c.

y

(

(

x

d.

y

e.

x

x

(0, -1)

y

f.

y

(-1, 0)

x

x

x

CONSTRUCTING VIABLE ARGUMENTS

To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.

(

(

(1,- 3 22

( 1 , -1 22

Communicate Your Answer

2. How can you use the unit circle to define the trigonometric functions of any angle?

3. For which angles are each function undefined? Explain your reasoning.

a. tangent

b. cotangent

c. secant

d. cosecant

Section 9.3 Trigonometric Functions of Any Angle 477

9.3 Lesson

Core Vocabulary

unit circle, p. 479 quadrantal angle, p. 479 reference angle, p. 480 Previous circle radius Pythagorean Theorem

What You Will Learn

Evaluate trigonometric functions of any angle. Find and use reference angles to evaluate trigonometric functions.

Trigonometric Functions of Any Angle

You can generalize the right-triangle definitions of trigonometric functions so that they apply to any angle in standard position.

Core Concept

General Definitions of Trigonometric Functions

Let be an angle in standard position, and let (x, y)

y

be the point where the terminal side of intersects

the circle x2 + y2 = r2. The six trigonometric

functions of are defined as shown.

(x, y)

sin = --yr

csc = --yr, y 0

r

x

cos = --xr

sec = --xr, x 0

tan = --xy, x 0

cot = --yx, y 0

These functions are sometimes called circular functions.

Evaluating Trigonometric Functions Given a Point

Let (-4, 3) be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of .

SOLUTION

y

(-4, 3)

r

Use the Pythagorean Theorem to find the length of r.

x

r = -- x2 + y2

= -- (-4)2 + 32

--

= 25

= 5

Using x = -4, y = 3, and r = 5, the values of the six trigonometric functions of are:

sin = --yr = --53

csc = --yr = --35

cos = --xr = ---54

sec = --xr = ---45

tan = --yx = ---43

cot = --xy = ---34

478 Chapter 9 Trigonometric Ratios and Functions

ANOTHER WAY

The general circle x2 + y2 = r2 can also be used to find the six trigonometric functions of . The terminal side of intersects the circle at (0, -r). So,

sin = --yr = -- -rr = -1.

The other functions can be evaluated similarly.

Core Concept

The Unit Circle

y

The circle x2 + y2 = 1, which has center (0, 0)

and radius 1, is called the unit circle. The values

of sin and cos are simply the y-coordinate and

x-coordinate, respectively, of the point where the

terminal side of intersects the unit circle.

x

sin = --yr = --1y = y

r = 1 (x, y)

cos = --xr = --1x = x

It is convenient to use the unit circle to find trigonometric functions of quadrantal angles. A quadrantal angle is an angle in standard position whose terminal side lies on an axis. The measure of a quadrantal angle is always a multiple of 90?, or --2 radians.

Using the Unit Circle

Use the unit circle to evaluate the six trigonometric functions of = 270?.

SOLUTION

y

Step 1 Draw a unit circle with the angle = 270? in

standard position.

Step 2 Identify the point where the terminal side

of intersects the unit circle. The terminal

x

side of intersects the unit circle at (0, -1).

Step 3 Find the values of the six trigonometric functions. Let x = 0 and y = -1 to evaluate the trigonometric functions.

(0, -1)

sin = --yr = -- -11 = -1

csc = --yr = -- -11 = -1

cos = --xr = --10 = 0

sec = --xr = --01 undefined

tan = --yx = -- -01 undefined

cot = --xy = -- -01 = 0

Monitoring Progress

Help in English and Spanish at

Evaluate the six trigonometric functions of .

1.

y

2. (-8, 15) y

3.

y

x

x

x

(3, -3)

(-5, -12)

4. Use the unit circle to evaluate the six trigonometric functions of = 180?.

Section 9.3 Trigonometric Functions of Any Angle 479

READING

The symbol is read as "theta prime."

Reference Angles

Core Concept

Reference Angle Relationships

Let be an angle in standard position. The reference angle for is the acute angle formed by the terminal side of and the x-axis. The relationship between and is shown below for nonquadrantal angles such that 90? < < 360? or, in radians, --2 < < 2.

Quadrant II

Quadrant III

Quadrant IV

y

y

y

x

x

x

Degrees: = 180? - Degrees: = - 180? Degrees: = 360? -

Radians: = -

Radians: = -

Radians: = 2 -

Finding Reference Angles

Find the reference angle for (a) = -- 53 and (b) = -130?.

SOLUTION

a. The terminal side of lies in Quadrant IV. So,

y

= 2 - -- 53 = --3 . The figure at the right shows

x

= -- 53 and = --3 .

b. Note that is coterminal with 230?, whose terminal side

lies in Quadrant III. So, = 230? - 180? = 50?. The

figure at the left shows = -130? and = 50?.

y

x

Reference angles allow you to evaluate a trigonometric function for any angle . The sign of the trigonometric function value depends on the quadrant in which lies.

Core Concept

Evaluating Trigonometric Functions

Use these steps to evaluate a trigonometric function for any angle :

Step 1 Find the reference angle .

Step 2 Evaluate the trigonometric function for .

Step 3

Determine the sign of the trigonometric function value from the quadrant in which lies.

Signs of Function Values

Quadrant II sin , csc : + cos , sec : - tan , cot : -

y Quadrant I sin , csc : + cos , sec : + tan , cot : +

Quadrant III sin , csc : -

cos , sec : -

tan , cot : +

Quadrant IV x sin , csc : -

cos , sec : +

tan , cot : -

480 Chapter 9 Trigonometric Ratios and Functions

INTERPRETING MODELS

This model neglects air resistance and assumes that the projectile's starting and ending heights are the same.

Using Reference Angles to Evaluate Functions Evaluate (a) tan(-240?) and (b) csc -- 176.

SOLUTION

a. The angle -240? is coterminal with 120?. The reference angle is = 180? - 120? = 60?. The tangent function is negative in Quadrant II, so

--

tan(-240?) = -tan 60? = -3.

b. The angle -- 176 is coterminal with -- 56. The reference angle is = - -- 56 = --6 . The cosecant function is positive in Quadrant II, so csc -- 176 = csc --6 = 2.

y

= 60?

x

= -240?

=

6

y

x

=176

Solving a Real-Life Problem

The horizontal distance d (in feet) traveled by a projectile launched at an angle and with an initial speed v (in feet per second) is given by

d = -- 3v22 sin 2.

Model for horizontal distance

Estimate the horizontal distance traveled by a golf ball

that is hit at an angle of 50? with an initial speed of

105 feet per second.

50?

SOLUTION

Note that the golf ball is launched at an angle of = 50? with initial speed of v = 105 feet per second.

d = -- 3v22 sin 2

= -- 130252 sin(2 50?)

Write model for horizontal distance. Substitute 105 for v and 50? for .

339

Use a calculator.

The golf ball travels a horizontal distance of about 339 feet.

Monitoring Progress

Help in English and Spanish at

Sketch the angle. Then find its reference angle.

5. 210?

6. -260?

7. -- -97

Evaluate the function without using a calculator.

8. -- 154

9. cos(-210?)

10. sec -- 114

11. Use the model given in Example 5 to estimate the horizontal distance traveled by a track and field long jumper who jumps at an angle of 20? and with an initial speed of 27 feet per second.

Section 9.3 Trigonometric Functions of Any Angle 481

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download