Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate

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

Over Lesson 4-2

Write 62.937˚ in DMS form.

A. 62°54'13"

B. 63°22'2"

C. 62°54'2"

D. 62°56'13.2"

Over Lesson 4-2

Write 96°42'16'' in decimal degree form to the nearest thousandth.

A. 96.704o

B. 96.422o

C. 96.348o

D. 96.259o

Over Lesson 4-2

Write 135º in radians as a multiple of π.

A.

B.

C.

D.

Over Lesson 4-2

A. 240o

B. –60o

C. –120o

D. –240o

Write in degrees.

Over Lesson 4-2

Find the length of the intercepted arc witha central angle of 60° in a circle with a radius of15 centimeters. Round to the nearest tenth.

A. 7.9 cm

B. 14.3 cm

C. 15.7 cm

D. 19.5 cm

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.

• quadrantal angle

• reference angle

• unit circle

• circular function

• periodic function

• period

Evaluate Trigonometric Functions Given a Point

Let (–4, 3) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ.

Pythagorean Theorem

x = –4 and y = 3

Use x = –4, y = 3, and r = 5 to write the six trigonometric ratios.

Take the positive square root.

Evaluate Trigonometric Functions Given a Point

Answer:

Let (–3, 6) be a point on the terminal side of an angle Ө in standard position. Find the exact values of the six trigonometric functions of Ө.

A.

B.

C.

D.

Evaluate Trigonometric Functions of Quadrantal Angles

A. Find the exact value of cos π. If not defined, write undefined.

The terminal side of π 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.


Answer: –1

x = –1 and r = 1

Cosine function


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.


Answer: undefined

y = 1 and x = 0

Tangent function


C. Find the exact value of . If not defined, write undefined.

The terminal side of in standard position lies

on the negative y-axis. The point (0, –1) is convenient

because r = 1.


Answer: 0

x = 0 and y = –1

Cotangent function

A. –1

B. 0

C. 1

D. undefined

Find the exact value of sec If not defined, write undefined.

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°

Find Reference Angles

Answer:

The terminal side of lies in Quadrant II. Therefore,

its reference angle is .

B. Sketch . Then find its reference angle.

Find the reference angle for a 520o angle.

A. 20°

B. 70°

C. 160°

D. 200°

Use Reference Angles to Find Trigonometric Values

A. Find the exact value of .

Because the terminal side of lies in Quadrant III, the

reference angle


Answer:

In Quadrant III, sin θ is negative.


B. Find the exact value of tan 150º.

Because the terminal side of θ lies in Quadrant II, the reference angle θ' is 180o – 150o or 30o.


Answer:

tan 150° = –tan 30° In Quadrant II, tan θ is negative.

tan 30°


C. Find the exact value of .

A coterminal angle of which lies in

Quadrant IV. So, the reference angle

Because cosine and secant are reciprocal functions

and cos θ is positive in Quadrant IV, it follows that

sec θ is also positive in Quadrant IV.


In Quadrant IV, sec θ is positive.


Answer:

CHECK You can check your answer by using a graphing calculator.

A.

B.

C.

D.

Find the exact value of cos .

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 θ. You know that sec θ is positive and sin θ is positive, so θ must lie in Quadrant I. This means that both x and y are positive.

Let , where sin θ > 0. Find the exact

values of the remaining five trigonometric

functions of θ.


Because sec =

and x = 5 to find y.

Take the positive square root.

Pythagorean Theorem

r = and x = 5


Use x = 5, y = 2, and r = to write the other five trigonometric ratios.


Answer:

Let csc θ = –3, tan θ < 0. Find the exact values o the five remaining trigonometric functions of θ.

A.

B.

C.

D.

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.

Find Coordinates Given a Radius and an Angle


Cosine ratio

= 150° and r = 10

cos 150° = –cos 30°

Solve for x.


Sin ratio

θ = 150° and r = 10

sin 150° = sin 30°

Solve for y.5 = y


Answer: The exact coordinates of D are .

The object is about 8.66 inches to the left of

the pivot point and 5 inches above the pivot

point.

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

Find Trigonometric Values Using the Unit Circle

Definition of sin tsin t = y

Answer:

A. Find the exact value of . If undefined,

write undefined.

corresponds to the point (x, y) = on

the unit circle.

y = . sin


Answer:

cos t = x Definition of cos t

cos

corresponds to the point (x, y) = on the

unit circle.

B. Find the exact value of . If undefined,

write undefined.


Definition of tan t.

C. Find the exact value of . If undefined,

write undefined.


Simplify.

Answer:


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

Definition of sec t

x = 0 when t = 270°

A.

B.

C.

D.

Find the exact value of tan . If undefined, write undefined.

Use the Periodic Nature of Circular Functions

cos t = x and x =

A. Find the exact value of .

Rewrite as the sum of a

number and 2π.

+ 2π map to the same

point (x, y) = on the

unit circle.


Answer:

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.


= sin 60o 60o and 60o

+ 360o(–1) map to the

same point

(x, y) =

on the unit

circle.


= sin t = y

and y = when t =

60o.

Answer:


C. Find the exact value of .

Rewrite as the sum of a

number and 2 and an integer

multiple of π.

map to the

same point (x, y) =

on the unit circle.


Answer:

A. 1

B. –1

C.

D.

Find the exact value of cos

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Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4-2) Then/Now New Vocabulary Key Concept: Trigonometric Functions of Any Angle Example 1: Evaluate