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Five-Minute Check (over Lesson 4-2) Then/Now New

Vocabulary Key Concept Trigonometric Functions

of Any Angle Example 1 Evaluate Trigonometric

Functions Given a Point Key Concept Common

Quadrantal Angles Example 2 Evaluate

Trigonometric Functions of Quadrantal Angles Key

Concept Reference Angle Rules Example 3 Find

Reference Angles Key Concept Evaluating

Trigonometric Functions of Any Angle Example 4

Use Reference Angles to Find Trigonometric

Values Example 5 Use One Trigonometric Value to

Find Others Example 6 Real-World Example Find

Coordinates Given a Radius and an Angle Key

Concept Trigonometric Functions on the Unit

Circle Example 7 Find Trigonometric Values

Using the Unit Circle Key Concept Periodic

Functions Example 8 Use the Periodic Nature of

Circular Functions

5Minute Check 1

Write 62.937 in DMS form.

A. 6254'13" B. 6322'2" C. 6254'2" D. 6256'13.2

"

5Minute Check 2

Write 9642'16'' in decimal degree form to the

nearest thousandth.

A. 96.704o B. 96.422o C. 96.348o D. 96.259o

5Minute Check 3

Write 135º in radians as a multiple of p.

5Minute Check 4

A. 240o B. 60o C. 120o D. 240o

5Minute Check 5

Find the length of the intercepted arc with a

central angle of 60 in a circle with a radius

of 15 centimeters. Round to the nearest tenth.

A. 7.9 cm B. 14.3 cm C. 15.7 cm D. 19.5 cm

Then/Now

You found values of trigonometric functions for

acute angles using ratios in right triangles.

(Lesson 4-1)

- Find values of trigonometric functions for any

angle. - Find values of trigonometric functions using the

unit circle.

Vocabulary

- quadrantal angle
- reference angle
- unit circle
- circular function
- periodic function
- period

Key Concept 1

Example 1

Evaluate Trigonometric Functions Given a Point

Let (4, 3) be a point on the terminal side of an

angle ? in standard position. Find the exact

values of the six trigonometric functions of ?.

Use x 4, y 3, and r 5 to write the six

trigonometric ratios.

Example 1

Evaluate Trigonometric Functions Given a Point

Answer

Example 1

Let (3, 6) be a point on the terminal side of an

angle ? in standard position. Find the exact

values of the six trigonometric functions of ?.

Key Concept 2

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

A. Find the exact value of cos p. If not defined,

write undefined.

The terminal side of p in standard position lies

on the negative x-axis. Choose a point P on the

terminal side of the angle. A convenient point is

(1, 0) because r 1.

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

Answer 1

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

B. Find the exact value of tan 450. If not

defined, write undefined.

The terminal side of 450 in standard position

lies on the positive y-axis. Choose a point P(0,

1) on the terminal side of the angle because r

1.

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

Answer undefined

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

Example 2

Evaluate Trigonometric Functions of Quadrantal

Angles

Cotangent function

Answer 0

Example 2

A. 1 B. 0 C. 1 D. undefined

Key Concept 3

Example 3

Find Reference Angles

A. Sketch 150. Then find its reference angle.

A coterminal angle is 150 360 or 210. The

terminal side of 210 lies in Quadrant III.

Therefore, its reference angle is 210 180 or

30.

Answer 30

Example 3

Find Reference Angles

Example 3

Find the reference angle for a 520o angle.

A. 20 B. 70 C. 160 D. 200

Key Concept 4

Example 4

Use Reference Angles to Find Trigonometric Values

Example 4

Use Reference Angles to Find Trigonometric Values

Example 4

Use Reference Angles to Find Trigonometric Values

B. Find the exact value of tan 150º.

Because the terminal side of ? lies in Quadrant

II, the reference angle ?' is 180o 150o or 30o.

Example 4

Use Reference Angles to Find Trigonometric Values

tan 150 tan 30 In Quadrant II, tan ? is

negative.

Example 4

Use Reference Angles to Find Trigonometric Values

Example 4

Use Reference Angles to Find Trigonometric Values

Example 4

Use Reference Angles to Find Trigonometric Values

CHECK You can check your answer by using a

graphing calculator.

Example 4

Example 5

Use One Trigonometric Value to Find Others

To find the other function values, you must find

the coordinates of a point on the terminal side

of ?. You know that sec ? is positive and sin ?

is positive, so ? must lie in Quadrant I. This

means that both x and y are positive.

Example 5

Use One Trigonometric Value to Find Others

Example 5

Use One Trigonometric Value to Find Others

Example 5

Use One Trigonometric Value to Find Others

Example 5

Let csc ? 3, tan ? lt 0. Find the exact values

o the five remaining trigonometric functions of ?.

Example 6

Find Coordinates Given a Radius and an Angle

ROBOTICS A student programmed a 10-inch long

robotic arm to pick up an object at point C and

rotate through an angle of 150 in order to

release it into a container at point D. Find the

position of the object at point D, relative to

the pivot point O.

Example 6

Find Coordinates Given a Radius and an Angle

Example 6

Find Coordinates Given a Radius and an Angle

Example 6

Find Coordinates Given a Radius and an Angle

Example 6

CLOCK TOWER A 4-foot long minute hand on a clock

on a bell tower shows a time of 15 minutes past

the hour. What is the new position of the end of

the minute hand relative to the pivot point at 5

minutes before the next hour?

A. 6 feet left and 3.5 feet above the pivot

point B. 3.4 feet left and 2 feet above the pivot

point C. 3.4 feet left and 6 feet above the pivot

point D. 2 feet left and 3.5 feet above the pivot

point

Key Concept 7

Example 7

Find Trigonometric Values Using the Unit Circle

Example 7

Find Trigonometric Values Using the Unit Circle

cos t x Definition of cos t

Example 7

Find Trigonometric Values Using the Unit Circle

Example 7

Find Trigonometric Values Using the Unit Circle

Example 7

Find Trigonometric Values Using the Unit Circle

D. Find the exact value of sec 270. If

undefined, write undefined.

270 corresponds to the point (x, y) (0, 1) on

the unit circle.

Therefore, sec 270 is undefined.

Answer undefined

Example 7

Key Concept 8

Example 8

Use the Periodic Nature of Circular Functions

Example 8

Use the Periodic Nature of Circular Functions

Example 8

Use the Periodic Nature of Circular Functions

B. Find the exact value of sin(300).

sin (300o) sin (60o 360o(1)) Rewrite 300o

as the sum of a number and an integer multiple of

360o.

Example 8

Use the Periodic Nature of Circular Functions

Example 8

Use the Periodic Nature of Circular Functions

Example 8

Use the Periodic Nature of Circular Functions

Example 8

End of the Lesson