4-1 Study Guide and Intervention · Chapter 4 7 Glencoe Precalculus 4-1 Practice Right Triangle...

NAME _____________________________________________ DATE ____________________________ PERIOD _____________

Chapter 4 5 Glencoe Precalculus

4-1 Study Guide and Intervention Right Triangle Trigonometry

Values of Trigonometric Ratios The side lengths of a right triangle and a reference angle θ can be used to form six

trigonometric ratios that define the trigonometric functions known as sine, cosine, and tangent. The cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent ratios, respectively. Therefore, they are known as

reciprocal functions.

Let θ be an acute angle in a right triangle and the abbreviations opp, adj, and hyp refer to the lengths of the side opposite θ, the side adjacent to θ, and the hypotenuse, respectively.

Then the six trigonometric functions of θ are defined as follows.

sine (θ) = sin θ = opp

hyp cosine (θ) = cos θ =

adj

hyp tangent (θ) = tan θ =

opp

adj

cosecant (θ) = csc θ = hyp

opp secant (θ) = sec θ =

hyp

adj cotangent (θ) = cot θ =

adj

opp

Example: Find the exact values of the six trigonometric functions of θ.

Use the Pythagorean Theorem to determine the length of the hypotenuse.

152+32= 𝑐2 a = 15, b = 3

234 = 𝑐2 Simplify.

c = √234 or 3√26 Take the positive square root.

sin θ = opp

hyp or

3

3√26 or

√26

26 cos θ =

adj

hyp or

15

3√26 or

√265

26 tan θ =

opp

adj or

3

15 or

1

5

csc θ = hyp

opp or

3√26

3 or

26

√26 sec θ =

hyp

adj or

3√26

15 or

√26

5 cot θ =

adj

opp or

15

3 or 5

Exercises

Find the exact values of the six trigonometric functions of θ.

1. 2.

Use the given trigonometric function value of the acute angle θ to find the exact values of the five remaining

trigonometric function values of θ.

3. sin 𝜃 = 3

7 4. sec 𝜃 =

8

5

sin θ = 𝟕

𝟏𝟗, cos θ =

𝟐√𝟕𝟖

𝟏𝟗, tan θ =

𝟕√𝟕𝟖

𝟏𝟓𝟔, csc θ =

𝟏𝟗

𝟕,

sin θ = 𝟐√𝟓

𝟓, cos θ =

√𝟓

𝟓, tan θ = 2, sec θ =

𝟏𝟗√𝟕𝟖

𝟏𝟓𝟔, cot θ =

𝟐√𝟕𝟖

𝟕

csc θ = √𝟓

𝟐, sec θ = √𝟓, cot θ =

𝟏

𝟐

cos θ = 𝟐√𝟏𝟎

𝟕, tan θ =

𝟑√𝟏𝟎

𝟐𝟎, sin θ =

√𝟑𝟗

𝟖, cos θ =

𝟓

𝟖, tan θ =

√𝟑𝟗

𝟓,

csc θ = 𝟕

𝟑, sec θ =

𝟕√𝟏𝟎

𝟐𝟎, cot θ =

𝟐√𝟏𝟎

𝟑 csc θ =

𝟖√𝟑𝟗

𝟑𝟗, cot θ =

𝟓√𝟑𝟗

𝟑𝟗

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-1 Study Guide and Intervention (continued)

Right Triangle Trigonometry

Solving Right Triangles To solve a right triangle means to find the measures of all of the angles and sides of the triangle. When the trigonometric value of an acute angle is known, the inverse of the trigonometric function can be used to find the

measure of the angle.

Example: Solve △ABC. Round side measures to the nearest tenth and angle measures

to the nearest degree.

Because two lengths are given, you can use the Pythagorean Theorem to find that a is equal

to √825 or about 28.7. Find the measure of ∠ A using the cosine function.

cos θ = adj

hyp Cosine function

cos A = 20

35 Substitute b = 20 and c = 35.

A = cos−1 20

35 Definition of inverse cosine

A = 55.15009542 Use a calculator

Because A is now known, you can find B by subtracting A from 90°.

55.15 + B = 90 Angles A and B are complementary.

B = 34.85° Subtract.

Therefore, a ≈ 28.7, A ≈ 55°, and B ≈ 35°.

Exercises Find the value of x. Round to the nearest tenth if necessary.

1. 2.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

3. 4.

Trigonometric Function Inverse Trigonometric Function

y = sin x y = cos x y = tan x

x = sin−1 𝑦 or 𝑥 = arcsin y x = cos−1 𝑦 or 𝑥 = arccos y x = tan−1 𝑦 or 𝑥 = arctan y

about 17.3 about 18.9 r = 9.5, A = 41°, S = 17.5°, b = 10.4, R = 72.5° c = 13.7

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-1 Practice Right Triangle Trigonometry

Find the exact values of the six trigonometric functions of θ.

1. 2.

Find the value of x. Round to the nearest tenth, if necessary.

3. 4.

5. On a college campus, the library is 80 yards due east of the dormitory and the recreation center is due north of the library. The college is constructing a sidewalk from the dormitory to the recreation center. The sidewalk will be at a

56° angle with the current sidewalk between the dormitory and the library. To the nearest yard, how long will the new

sidewalk be?

6. If cot A = 8, find the exact values of the remaining trigonometric functions for the acute angle A.

Find the measure of angle θ. Round to the nearest degree, if necessary.

7. 8.

Solve each triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.

9. 10.

11. SWIMMING The swimming pool at Perris Hill Plunge is 50 feet long and 25 feet wide. If the bottom of the pool is

slanted so that the water depth is 3 feet at the shallow end and 15 feet at the deep end, what is the angle of elevation at

the bottom of the pool?

sin θ = 𝟑

𝟖, cos θ =

√𝟓𝟓

𝟖, sin θ =

𝟐𝟒

𝟐𝟓,

tan θ = 𝟑√𝟓𝟓

𝟓𝟓, csc θ =

𝟖

𝟑, cos θ =

𝟕

𝟐𝟓,

sec θ = 𝟖√𝟓𝟓

𝟓𝟓, cot θ =

√𝟓𝟓

𝟑 tan θ =

𝟐𝟒

𝟕, csc θ =

𝟐𝟓

𝟐𝟒,

sec θ = 𝟐𝟓

𝟕, cot θ =

𝟕

𝟐𝟒

35.3 9.4

143 yd

sin A = √𝟔𝟓

𝟔𝟓, cos A =

𝟖√𝟔𝟓

𝟔𝟓, tan A =

𝟏

𝟖, sec A =

√𝟔𝟓

𝟖, csc A = √𝟔𝟓

42° 55° a ≈ 5.2, c ≈ 7.3, b ≈ 13.0, A ≈ 16°, B = 68° B ≈ 74°

about 13.5°

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-2 Study Guide and Intervention Degrees and Radians

Angles and Their Measures One complete rotation can be represented by 360° or 2π radians. Thus, the following

formulas can be used to relate degree and radian measures.

Degree/Radian Coversion Rules

1° = π

180 radians 1 radian = (

180

𝜋) °

If two angles have the same initial and terminal sides, but different measures, they are called coterminal angles.

Example: Write each degree measure in radians as a multiple of π and each radian measure in degrees.

a. 36°

36° = 36° (𝜋 radians

180°) Multiply by

π radians

180°

= π

5 radians or

π

5 Simplify

b. −𝟏𝟕𝛑

𝟑

– 17π

3 = −

17π

3 radians Multiply by

180°

π radians

= − 17π

3 radians (

180°

𝜋 radians) = −1020° Simplify

Exercises

Write each degree measure in radians as a multiple of π and each radian measure in degrees.

1. −250° 2. 6° 3. −145°

4. 870° 5. 18° 6. −820°

7. 4π 8. 13π

30 9. −1

10. 3π

16 11. −2.56 12. −

7π

9

Identify all angles that are coterminal with the given angle.

13. − π

2 14. 135° 15.

5π

3

− 𝟐𝟓𝝅

𝟏𝟖

𝝅

𝟑𝟎 −

𝟐𝟗𝝅

𝟑𝟔

𝟐𝟗𝝅

𝟔

𝝅

𝟏𝟎 −

𝟒𝟏𝝅

𝟗

720° 78° −57.3°

33.75° −146.7° −140°

− 𝝅

𝟐 + 2nπ 135° + 360n°

𝟓𝝅

𝟑 + 2nπ

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Degrees and Radians

Applications with Angle Measure The rate at which an object moves along a circular path is called its linear speed.

The rate at which the object rotates about a fixed point is called its angular speed.

Suppose an object moves at a constant speed along a circular path of radius r.

If s is the arc length traveled by the object during time t, then the object’s linear speed v is given by

V = 𝑠

𝑡,

If θ is the angle of rotation (in radians) through which the object moves during time t, then the angular speed ω of the object is given by

ω = 𝜃

𝑡 .

Example: Determine the angular speed and linear speed if 8.2 revolutions are completed in 3 seconds and the

distance from the center of rotation is 7 centimeters. Round to the nearest tenth.

The angle of rotation is 8.2 × 2π or 16.4π radians.

ω = θ

t Angular speed

= 16.4π

3 θ = 16.4π radians and t = 3 seconds

≈ 17.17403984 Use a calculator.

Therefore, the angular speed is about 17.2 radians per second.

The linear speed is 𝑟𝜃

𝑡.

V= 𝑠

𝑡, Linear speed

= 𝑟𝜃

𝑡 s = rθ

= 7(16.4𝜋)

3 r = 7 centimeters, θ = 16.4π radians, and t = 3 seconds

= 120.218278877 Use a calculator.

Therefore, the linear speed is about 120.2 centimeters per second.

Exercises Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the

rate of rotation.

1. ω = 2.7 rad/s 2. ω = 4

3π rad/hr

3. ω = 3

2π rad/min 4. V = 24.8 m/s, 120 rev/min

5. V = 118 ft/min , 3.6 rev/s 6. V = 256 in./h, 0.5 rev/min

25.8 rev/min 0.01 rev/min

𝟑

𝟒 rev/min 2 m

0.09 ft 1.4 in.

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-2 Practice Degrees and Radians

Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest

thousandth.

1. 28.955 2. –57.3278

3. 32 28' 10" 4. –73 14' 35"

Write each degree measure in radians as a multiple of π and each radian measure in degrees.

5. 25° 6. 130°

7. 3π

4 8.

5π

3

Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative

angle coterminal with the given angle.

9. 43° 10. − 7π

4

Find the length of the intercepted arc with the given central angle measure in a circle of the given radius. Round to

the nearest tenth.

11. 30°, r = 8 yd 12. 7π

6, r = 10 in.

Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the

rate of rotation.

13. ω = 4

5π rad/s 14. V = 32 m/s, 100 rev/min

15. On a game show, a contestant spins a wheel. The angular speed of the wheel was ω = π

3 radians per second. If the

wheel maintained this rate, what would be the rotation in revolutions per minute?

Find the area of each sector.

16. θ = π

6, r = 14 in. 17. θ =

7π

4, r = 4 m

28° 57′ 18" –57° 19′ 40.08" 32.469° –73.243°

𝟓𝛑

𝟑𝟔

𝟏𝟑𝛑

𝟏𝟖

135° 300°

43° + 360n° −𝟕𝛑

𝟒 + 2n𝛑

Sample answers: 403°, –317° Sample answers: 𝛑

𝟒, –

𝟏𝟓𝛑

𝟒

4.2 yd 36.7 in.

24 rev/min 3.06 m

10 rev/min

51.3 in2 44.0 m2

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-3 Study Guide and Intervention Trigonometric Functions on the Unit Circle

Trigonometric Functions of Any Angle The definitions of the six trigonometric functions may be extended to include

any angle as shown below.

Let θ be any angle in standard position and point P(x, y) be a point on the terminal side of θ.

Let r represent the nonzero distance from P to the origin. That is, let r = √𝑥2 + 𝑦2 ≠ 0.

Then the trigonometric functions of θ are as follows.

sin θ = 𝑦

𝑟 csc θ =

𝑟

𝑦, y ≠ 0

cos θ = 𝑥

𝑟 sec θ=

𝑟

𝑥, x ≠ 0

tan θ = 𝑦

𝑥, x ≠ 0 cot θ =

𝑥

𝑦, y ≠ 0

You can use the following steps to find the value of a trigonometric function of any angle θ.

1. Find the reference angle θ′.

2. Find the value of the trigonometric function for θ′.

3. Use the quadrant in which the terminal side of θ lies to determine the sign of the trigonometric function value of θ.

Example: Let (−9, 12) 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.

Use the values of x and y to find r.

r = √𝑥2 + 𝑦2 Pythagorean Theorem

= √(−9)2 + 122 x = –9 and y = 12

= √225 or 15 Take the positive square root.

Use x = –9, y = 12, and r = 15 to write the six trigonometric ratios.

sin θ = 𝑦

𝑟 =

12

15 or

4

5 cos θ =

𝑥

𝑟 =

−9

15 or −

3

5 tan θ =

𝑦

𝑥 =

12

−9 or −

4

3

csc θ = 𝑟

𝑦 =

15

12 or

5

4 sec θ =

𝑟

𝑥 =

15

−9 or −

5

3 cot θ =

𝑥

𝑦 =

−9

12 or −

3

4

Exercises

The given point lies on the terminal side of an angle θ in standard position. Find the values of the six trigonometric

functions of θ.

1. (2, –5) 2. (12, 4) 3. (–3, –8)

Find the exact value of each expression.

4. sin5π

3 5. csc 210° 6. cot (–315°)

sin θ = – 𝟓√𝟐𝟗

𝟐𝟗, cos θ =

𝟐√𝟐𝟗

𝟐𝟗, sin θ =

√𝟏𝟎

𝟏𝟎, cos θ =

𝟑√𝟏𝟎

𝟏𝟎, sin θ = –

𝟖√𝟕𝟑

𝟕𝟑, cos θ = –

𝟑√𝟕𝟑

𝟕𝟑,

tan θ = – 𝟓

𝟐, csc θ = –

√𝟐𝟗

𝟓, tan θ =

𝟏

𝟑, csc θ = √𝟏𝟎, tan θ =

𝟖

𝟑, csc θ = –

√𝟕𝟑

𝟖,

sec θ = √𝟐𝟗

𝟐, cot θ = −

𝟐

𝟓 sec θ =

√𝟏𝟎

𝟑, cot θ = 3 sec θ = –

√𝟕𝟑

𝟑, cot θ =

𝟑

𝟖

– √𝟑

𝟐 – 2 1

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-3 Study Guide and Intervention (continued) Trigonometric Functions on the Unit Circle

Trigonometric Functions on the Unit Circle You can use the unit circle to find the values of the six trigonometric

functions for θ. The relationships between θ and the point P(x, y) on the unit circle are shown below.

Let t be any real number on a number line and let P(x, y) be the point on t when the number line is wrapped onto the unit circle. Then the trigonometric functions of t are as follows.

sin t = y cos t = x tan t = 𝑦

𝑥, x ≠ 0

csc t = 1

𝑦, y ≠ 0 sec t =

1

𝑥, x ≠ 0 cot t =

𝑥

𝑦, y ≠ 0

Therefore, the coordinates of P corresponding to the angle t can be written as P(cos t, sin t).

Example: Find the exact value of tan 𝟓𝛑

𝟑. If undefined, write undefined.

5π

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

1

2, −

√3

2) on the unit circle.

tan t = 𝑦

𝑥 Definition of tan t

tan 5𝜋

3 =

−√3

21

2

x = 1

2 and y = −

1

2 when t =

5𝜋

3

tan 5𝜋

3 = −√3 Simplify.

Exercises Find the exact value of each expression. If undefined, write undefined.

1. tan π

2 2. sec −

3π

4

3. cos 7π

6 4. sin

5π

4

5. cot 4π

3 6. csc −

5π

3

7. tan −60° 8. cot 270°

undefined −√𝟐

− √𝟑

𝟐 −

√𝟐

𝟐

√𝟑

𝟑

𝟐√𝟑

𝟑

−√𝟑 0

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-3 Practice Trigonometric Functions on the Unit Circle

The given point lies on the terminal side of an angle θ in standard position.

Find the values of the six trigonometric functions of θ.

1. (–1, 5) 2. (7, 0) 3. (–3, –4)

4. (1, –2) 5. (–3, 1) 6. (2, –4)

Sketch each angle. Then find its reference angle.

7. 330° 8. – 3π

4 9.

7π

6

10. 7π

4 11. 135° 12. –

π

3

Find the exact value of each expression. If undefined, write undefined.

13. csc 90° 14. tan 270° 15. sin (−90°)

16. cos 3π

2 17. sec (−

𝜋

4) 18. cot

5π

6

19. PENDULUMS The angle made by the swing of a pendulum and its vertical resting position can be modeled by

θ = 3 cos πt, where t is time measured in seconds and θ is measured in radians. What is the angle made by the

pendulum after 6 seconds?

sin θ = 𝟓√𝟐𝟔

𝟐𝟔, cos θ = −

√𝟐𝟔

𝟐𝟔, sin θ = 0, cos θ = 1, sin θ = −

𝟒

𝟓, cos θ = −

𝟑

𝟓,

tan θ = –5, csc θ = √𝟐𝟔

𝟓, tan θ = 0, csc θ = undefined, tan θ =

𝟒

𝟑, csc θ = −

𝟓

𝟒,

sec θ = −√𝟐𝟔, cot θ = −𝟏

𝟓 sec θ = 1, cot θ = undefined, sec θ = −

𝟓

𝟑, cot θ =

𝟑

𝟒,

sin θ = −𝟐√𝟓

𝟓, cos θ =

√𝟓

𝟓, sin θ =

√𝟏𝟎

𝟏𝟎, cos θ = −

𝟑√𝟏𝟎

𝟏𝟎, sin θ = −

𝟐√𝟓

𝟓, cos θ =

√𝟓

𝟓,

tan θ = –2, csc θ = −√𝟓

𝟐, tan θ = −

𝟏

𝟑, csc θ = √𝟏𝟎, tan θ = –2, csc θ = −

√𝟓

𝟐,

sec θ = √𝟓, cot θ = −𝟏

𝟐 sec θ = −

√𝟏𝟎

𝟑, cot θ = –3 sec θ = √𝟓, cot θ = −

𝟏

𝟐,

30° 𝝅

𝟒

𝝅

𝟔

𝝅

𝟒 45°

𝝅

𝟑

1 undefined –1

0 √𝟐 −√𝟑 3 rad

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-4 Study Guide and Intervention Graphing Sine and Cosine Functions

Transformations of Sine and Cosine Functions A sinusoid is a transformation of the graph of the sine function. The

general form of the sinusoidal functions sine and cosine are y = a sin (bx + c) + d or y = a cos (bx + c) + d. The graphs of y = a sin (bx + c) + d and y = a cos (bx + c) + d have the following characteristics.

Amplitude = |a| Period = 2π

|𝑏| Frequency =

|𝑏|

2π or

1

period

Phase Shift = c

|b| Vertical Shift = d Midline y = d

Example: State the amplitude, period, frequency, phase shift, and vertical shift of y = –2 cos (𝒙

𝟒−

𝝅

𝟑) + 2.

Then graph two periods of the function.

Amplitude = |a| = |–2| = 2

Period = 2π

|𝑏| =

2π

|1

4| or 8π

Frequency = |𝑏|

2π =

|1

4|

2π or

1

8π

Phase Shift = – 𝑐

|𝑏| =

−(−π

3)

|1

4|

or 4π

3

Vertical Shift = d or 2

Exercises State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of

the function.

1. y = 3 sin (2x + π) 2. y = cos (𝑥 − 𝜋

3) + 2

State the frequency and midline of each function.

3. y = 3 sin(1

2𝑥 +

4𝜋

3) + 1 4. y = cos (3x) – 2

Write a sine function with the given characteristics.

5. amplitude = 2, period = 4, phase shift = 1

2, vertical shift = 4

6. amplitude = 1.2, phase shift = 3π


3, π, 𝟏

𝝅, –

𝝅

𝟐, 0 1, 2π,

𝟏

𝟐𝝅,

𝝅

𝟑, 2

𝟏

𝟒𝝅; y = 1

𝟑

𝟐𝝅; y = –2

Sample answer: y = 2 sin (𝝅

𝟐𝒙 +

𝝅

𝟒) + 4

Sample answer: y = 1.2 sin (𝒙 −𝟑𝝅

𝟐) + 1

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Graphing Sine and Cosine Functions

Applications of Sinusoidal Functions You can use sinusoidal functions to solve certain application problems.

Example: The table shows the average monthly temperatures for Ann Arbor,

Michigan, in degrees Fahrenheit (°F). Write a sinusoidal function that models

the average monthly temperatures as a function of time x, where x = 1

represents January.

The data can be modeled by a sinusoidal function of the form y = a sin (bx + c) + d.

Find the maximum M and minimum m values of the data, and use these values to

find a, b, c, and d.

𝛼 = 1

2(𝑀 − 𝑚) Amplitude formula

= 1

2(84 − 30) or 27 M = 84 and m = 30

D = 1

2 (𝑀 + 𝑚) Vertical shift formula

= 1

2(84 + 30) or 57 M = 84 and m = 30

Period = 2(𝑥max − 𝑥min) 𝑥max = July or month 7 and

= 2(7 – 1) or 12 𝑥min = January or month 1

|𝑏| = 2π

12 or

π

6 |𝑏| =

2π

period

Phase shift = −c

|b| Phase shift formula

4 = −cπ

6

Phase shift = 4 and ⎪b⎥ = π

6

c = −2π

3 Solve for c.

Month Temperature

Jan 30°

Feb 34°

Mar 45°

Apr 59°

May 71°

June 80°

July 84°

Aug 81°

Sept 74°

Oct 62°

Nov 48°

Dec 35°

Therefore, y = 27 sin (𝜋

6𝑥 −

2𝜋

3) + 57 is one model for the average monthly temperature in Ann Arbor, Michigan.

Exercise

1. MUSEUM ATTENDANCE The table gives the number of visitors in thousands to a museum for each month.

Month Jan Feb Mar Apr May Jun Jul Aug

Visitors 10 8 11 15 24 30 32 29

a. Write a trigonometric function that models the monthly attendance at the museum using x = 1 to represent January.

b. According to your model, how many people should the museum expect to visit during October?

y = 12 sin (𝝅

𝟓𝒙 −

𝟗𝝅

𝟏𝟎) + 20

16,292 people

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-4 Practice Graphing Sine and Cosine Functions

Describe how the graphs of f(x) and g(x) are related. Then find the amplitude of g(x) and sketch two periods of

both functions on the same coordinate axes.

1. f(x) = sin x 2. f(x) = cos x

g(x) = 1

3 sin x g(x) = –

1

4 cos x

State the amplitude, period, frequency, phase shift, and vertical shift of each function. Then graph two periods of

the function.

3. y = 2 sin (𝑥 +𝜋

2) − 3 4. y =

1

2 cos (2x − π) + 2

Write a sinusoidal function with the given amplitude, period, phase shift, and vertical shift.

5. sine function: amplitude = 15, period = 4π, phase shift = π

2, vertical shift = –10

6. cosine function: amplitude = 2

3, period =

π

3, phase shift = –

π


7. MUSIC A piano tuner strikes a tuning fork note A above middle C and sets in motion vibrations that can be modeled

by y = 0.001 sin 880tπ. Find the amplitude and period of the function.

The graph of g(x) is the graph of The graph of g(x) is the graph of f(x) compressed vertically. The f(x) compressed vertically and

amplitude of g(x) is 𝟏

𝟑. reflected in the x-axis. The

amplitude of g(x) is 𝟏

𝟒.

amplitude = 2; period = 2𝝅; amplitude = 𝟏

𝟐; period = 𝛑;

frequency = 𝟏

𝟐𝝅; phase shift = −

𝝅

𝟐; frequency =

𝟏

𝝅; phase shift =

𝝅

𝟐;

vertical shift = –3 vertical shift = 2

y = ±15 sin ( 𝒙

𝟐−

𝝅

𝟒) – 10

y = ±𝟐

𝟑 cos (6x + 2𝛑) + 5

0.001; 𝟏

𝟒𝟒𝟎

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-5 Study Guide and Intervention Graphing Other Trigonometric Functions

Tangent and Reciprocal Functions You can use the same techniques you learned for graphing the sine and cosine

functions to graph the tangent function and the reciprocal trigonometric functions—cotangent, secant, and cosecant. The general form of the tangent function, which is similar to that of the sinusoidal functions, is

y = a tan (bx + c) + d,

where a produces a vertical stretch or compression, b affects the period, c produces a horizontal translation, and d

produces a vertical shift. The term amplitude does not apply to the tangent or cotangent functions because the heights of these functions are infinite.

Example: Locate the vertical asymptotes and sketch the graph of y = sec (2x + π) + 4.

The period of y = sec (2x + π) + 4 is 2𝜋

|𝑏| or π. Since c = π, the phase shift is −

𝜋

2. Therefore, the vertical asymptotes are

located every 𝜋

2 units at −

3𝜋

4, −

𝜋

4,

𝜋

4,

3𝜋

4, and

5𝜋

4. The vertical shift is 4. Create a table including the relative maximum and

minimum points for the period [− 𝜋

2,

3𝜋

2].

Functions Vertical

Asymptote Relative Minimum

Vertical Asymptote

Relative Maximum

Vertical Asymptote

Relative Minimum

Vertical Asymptote

y = sec x x = −𝜋

2 (0, 1) x =

𝜋

2 (π, –1) x =

3𝜋

2 (2π, 1) x =

5𝜋

2

y = sec (2x) x = −𝜋

4 (0, 1) x =

𝜋

4 (

𝜋

2, −1) x =

3𝜋

4 (π, 1) x =

5𝜋

4

y = sec (2x + π) x = −3𝜋

4 (−

𝜋

2, 1) x = −

𝜋

4 (0, –1) x =

𝜋

4 (

𝜋

2, 1) x =

3𝜋

4

y = sec (2x + π) + 4 x = −3𝜋

4 (−

𝜋

2, 5) x = −

𝜋

4 (0, 3) x =

𝜋

4 (

𝜋

2, 5) x =

3𝜋

4

Graph one cycle on the interval [− 𝜋

2,

3𝜋

2].

Then sketch one cycle to the left and right.

Exercises

Locate the vertical asymptotes and sketch the graph of each function.

1. y = 3 tan (4πx) 2. y = csc (2x – 3π)

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Graphing Other Trigonometric Functions

Damped Trigonometric Functions A damped trigonometric function is the product of a sinusoidal function of the

form y = sin bx or y = cos bx and another function y = f(x), called the damping factor. The graph of the product function is the oscillation of the sinusoidal function between the graphs of y = f(x) and y = –f(x). The resulting graph is called a

damped wave, and the reduction in the amplitude is known as damped oscillation. Damped oscillation can occur as x

approaches ±∞, as x approaches 0, or both.

Damped harmonic motion occurs when the amplitude of a function is damped due to friction over time. An object is in

damped harmonic motion when the amplitude is determined by the function a(t) = 𝑘𝑒−𝑐𝑡. For y = 𝑘𝑒−𝑐𝑡 sin ωt and

y = 𝑘𝑒−𝑐𝑡 cos ωt, where c > 0, k is the displacement, c is the damping constant, t is time, and ω is the period.

Example: Identify the damping factor f(x) of y = 𝟏

𝟐𝒙𝟐 sin 5x. Then use a graphing calculator to sketch the graphs

of f(x), −f(x), and the given function in the same viewing window.

Describe the behavior of the graph.

The function y = 1

2𝑥2 sin 5x is the product of the functions y =

1

2𝑥2 and y = sin 5x.

Therefore, the damping factor is f(x) = 1

2𝑥2. The amplitude of the function is

decreasing as x approaches zero.

Exercises

Identify the damping factor f(x) of each function. Then use a graphing calculator to sketch the graphs of f(x), −f(x),

and the given function in the same viewing window. Describe the behavior of the graph.

1. y = 3x sin 2x 2. y = 1

2 x cos 4x

3. A guitar string is plucked at a distance of 0.9 centimeter above its resting position and then released, causing vibration. The damping constant of the guitar string is 1.8, and the note produced has a frequency of 185 cycles per second.

a. Write a trigonometric function that models the motion of the string.

b. Determine the amount of time t that it takes the string to be damped so that –0.28 ≤ y ≤ 0.28.

f(x) = 3x; The amplitude of the function f(x) = 𝟏

𝟐x; The amplitude of the function

is decreasing as x approaches zero. is decreasing as x approaches zero.

y = 0.9e –1.8t cos 370πt

0.54 s

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-5 Practice Graphing Other Trigonometric Functions

Locate the vertical asymptotes, and sketch the graph of each function.

1. y = −3 tan x 2. y = −2 cot (2𝑥 + 𝜋

3)

3. y = csc x + 3 4. y = sec (𝑥

3 + 𝜋) – 1

Identify the damping factor f(x) of the function. Then use a graphing calculator to sketch the graphs of f(x), −f(x),

and the given function in the same viewing window. Describe the behavior of the graph.

5. y = 1

2x cos 2x 6. y = −

3

2x sin

𝜋𝑥

2

7. MUSIC A guitar string is plucked at a distance of 0.6 centimeter above its resting position and then released, causing vibration. The damping constant of the guitar string is 1.8, and the note produced has a frequency of 105 cycles per

second.

a. Write a trigonometric function that models the motion of the string.

b. Determine the amount of time t that it takes the string to be damped so that –0.24 ≤ y ≤ 0.24.

f(x) = 𝟏

𝟐𝒙; f(x) = −

𝟑

𝟐𝒙;

the amplitude of the function is the amplitude of the function is decreasing as x approaches 0 decreasing as x approaches 0

y = 0.6e–1.8t cos 210πt

0.5 s

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-6 Study Guide and Intervention Inverse Trigonometric Functions

Inverse Trigonometric Functions When restricted to a certain domain, the sine, cosine, and tangent functions have

inverse functions known as the arcsine, arccosine, and arctangent functions, respectively.

Inverse Trigonometric Functions

Inverse Sine of x y = sin−1 x or y = arcsin x if and only if

sin y = x, for –1 ≤ x ≤ 1 and –𝜋

2 ≤ y ≤

𝜋

2.

Inverse Cosine of x y = cos−1 x or y = arccos x if and only if cos y = x, for –1 ≤ x ≤ 1 and 0 ≤ y ≤ π.

Inverse Tangent of x y = tan−1 x or y = arctan x if and only if

tan y = x, for –∞ ≤ x ≤ ∞ and –𝜋

2 < y <

𝜋

2.

Examples: Find the exact value of each expression, if it exists.

1. 𝐬𝐢𝐧−𝟏 (−𝟏

𝟐)

Find a point on the unit circle in the interval [−𝜋

2,

𝜋

2] with a y-coordinate of −

1

2.

When t = −𝜋

6, sin t = −

1

2. Therefore, sin−1 (−

1

2) = –

𝜋

6.

2. 𝐜𝐨𝐬−𝟏 4

Because the domain of the inverse cosine function is [−1, 1] and 4 > 1, there is no angle with a cosine of 4. Therefore,

the value of cos−1 4 does not exist.

Exercises

Find the exact value of each expression, if it exists.

1. arctan 0 2. arcsin √3

2

3. cos−1 √2

2 4. tan−1 (−1)

Sketch the graph of the function.

5. y = arcsin 2x – 1 6. y = tan−1 (x – 1)

0 𝝅

𝟑

𝝅

𝟒 −

𝝅

𝟒

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



Inverse Trigonometric Functions

Compositions of Trigonometric Functions Because the domains of the trigonometric functions are restricted to

obtain the inverse trigonometric functions, the composition of a trigonometric function and its inverse does not follow the rules that you learned in Lesson 1-7. The properties that apply to trigonometric functions and their inverses are

summarized below.

Inverse Properties of Trigonometric Functions

f( 𝒇−𝟏(x)) = x If –1 ≤ x ≤ 1, then sin (sin−1 x) = x.

If –1 ≤ x ≤ 1, then cos (cos−1 x) = x.

If –∞ < x < ∞, then tan (tan−1 x) = x.

𝒇−𝟏(f(x)) = x

If –𝜋

2 ≤ x ≤

𝜋

2, then sin−1 (sin x) = x.

If 0 ≤ x ≤ π, then cos−1 (cos x) = x.

If –𝜋

2< x <

𝜋

2, then tan−1 (tan x) = x.

Example: Find the exact value of cos [𝐭𝐚𝐧−𝟏 −𝟒

𝟑].

To simplify the expression, let u = tan –1 (−4

3) , so tan u = −

4

3.

Because the tangent function is negative in Quadrants II and IV and the domain of the

inverse tangent function is restricted to Quadrants I and IV, u must lie in Quadrant IV.

Using the Pythagorean Theorem, you can find that the length of the hypotenuse is 5.

Now solve for cos u.

cos u = adj

hyp Cosine function

= 3

5 adj = 3 and hyp = 5

So, cos [tan−1 −4

3] =

3

5.

Exercises


1. sin (sin−1 −3

4) 2. cos−1 (cos

π

2)

3. tan−1 (tan 3π

2) 4. sin−1 (cos

π

6)

5. cos (arcsin 1

2) 6. tan (arcsin −

1

2)

7. cos (arccos 2) 8. cos (arctan −√3

3)

– 𝟑

𝟒

𝝅

𝟐

does not exist 𝝅

𝟑

√𝟑

𝟐 –

√𝟑

𝟑

does not exist √𝟑

𝟐

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-6 Practice Inverse Trigonometric Functions

Sketch the graph of each function.

1. y = arccos 3x 2. y = arctan x + 1

3. y = sin−1 3x 4. y = tan−1 3x


5. arcsin−1 (−√3

2) 6. cos −1 (cos

π

3)

7. tan (−3𝜋

2) 8. sin −1 (cos

π

3)

9. arctan (−√3

3) 10. arcsin (−

1

2)

11. tan (sin −1 1 − cos −1 1

2) 12. sin (arctan −

√3

3)

13. ART Hans purchased a painting that is 30 inches tall that will hang 8 inches above the fireplace. The top of the

fireplace is 55 inches from the floor.

a. Write a function modeling the maximum viewing angle θ for the distance d for Hans if his eye-level when sitting is

2.5 feet above the ground.

b. Determine the distance that corresponds to the maximum viewing angle.

Write each trigonometric expression as an algebraic expression of x.

14. sin (arccos x) 15. tan (sin−1 x)

– 𝝅

𝟑

𝝅

𝟑

undefined 𝝅

𝟔

– 𝝅

𝟔 –

𝝅

𝟔

√𝟑

𝟑 –

𝟏

𝟐

θ = 𝐭𝐚𝐧−𝟏 (𝟔𝟑

𝒅) – 𝐭𝐚𝐧−𝟏 (

𝟑𝟑

𝒅)

46.6 in.

√𝟏 − 𝒙𝟐 𝒙√𝟏− 𝒙𝟐

𝟏− 𝒙𝟐

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-7 Study Guide and Intervention The Law of Sines and the Law of Cosines

Solve Oblique Triangles The Law of Sines can be used to solve an oblique triangle when given the measures of two

angles and a nonincluded side (AAS), two angles and the included side (ASA), or two sides and a nonincluded angle (SSA).

The Law of Cosines can be used to solve an oblique triangle when given the measures of three sides (SSS) or the

measures of two sides and their included angle (SAS).

Example: Solve △ABC. Round side lengths to the nearest tenth and angle

measures to the nearest degree.

Because two angles are given,

C = 180° – (100° + 30°) or 50°.

Use the Law of Sines to find b and c.

sin 𝐴

𝑎 =

sin 𝐵

𝑏 Law of Sines

sin 𝐴

𝑎 =

sin 𝐶

𝑐

sin 30°

15 =

sin 100°

𝑏 Substitution

sin 30°

15 =

sin 50°

𝑐

b sin 30° = 15 sin 100° Cross products c sin 30° = 15 sin 50°

b = 15 sin 100°

sin 30° Divide each side by sin 30°. c =

15 sin 50°

sin 30°

b ≈ 29.5 Use a calculator. c ≈ 23.0

Therefore, b ≈ 29.5, c ≈ 23.0, and C = 50°.

Exercises

Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.

1. 2.

3. 4.

A = 38°, B = 22°, C = 120° b = 13.8, A = 45°, C = 35° B = 10°, b = 69.5, c = 136.8 C = 105°, b = 14.1, c = 19.3

NAME _____________________________________________ DATE ____________________________ PERIOD _____________



The Law of Sines and the Law of Cosines

Find Areas of Oblique Triangles When the measures of all three sides of an oblique triangle are known,

Heron’s Formula can be used to find the area of the triangle.

Heron’s Formula

If the measures of the sides of △ABC are a, b, and c, then the area of the triangle is

√𝑠 (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐),

where s = 1

2(a + b + c).

When two sides and the included angle of a triangle are known, the area is one-half the product of the lengths of the two

sides and the sine of the included angle.

Area of a Triangle Given SAS

Area = 1

2bc sin A

Area = 1

2ac sin B

Area = 1

2ab sin C

Example: Find the area of △XYZ to the nearest tenth.

The value of s is 1

2 (20 + 22 + 36) or 39.

Area = √𝑠(𝑠 − 𝑥)(𝑠 − 𝑦)(𝑠 − 𝑧) Heron’s Formula

= √39(39 − 20)(39 − 22)(39 − 36) s = 39, x = 20, y = 22, and z = 36

= √37,791 or about 194.4 in2 Simplify.

Exercises

Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.

1. △ABC if a = 14 ft, b = 9 ft, c = 8 ft 2. △FGH if f = 8 in., g = 9 in., h = 3 in.

3. △MNP if m = 3 yd, n = 4.6 yd, p = 5 yd 4. △XYZ if x = 8 cm, y = 12 cm, z = 13 cm

Find the area of each triangle to the nearest tenth.

5. △RST if R = 50°, s = 12 yd, t = 14 yd 6. △MNP if n = 14 ft, P = 110°, N = 25°

7. △DEF if d = 15 ft, E = 135°, f = 18 ft 8. △JKL if j = 4.3 m, l = 3.9 m, K = 82°

33.7 ft2 11.8 in2

6.8 yd2 47 cm2

64.3 yd2 154.1 ft2

95.5 ft2 8.3 m2

NAME _____________________________________________ DATE ____________________________ PERIOD _____________


4-7 Practice The Law of Sines and the Law of Cosines

Solve each triangle. Round to the nearest tenth if necessary.

1. 2.

3. 4.

5. STREET LIGHTING A lamp post tilts toward the Sun at a 2° angle from the vertical and casts a 25-foot shadow. The angle from the tip of the shadow

to the top of the lamp post is 45°. Find the length of the lamp post.

Use Heron’s Formula to find the area of each triangle. Round to the nearest tenth.

6. △ABC if a = 5 ft, b = 12 ft, c = 13 ft 7. △FGH if f = 11 in., g = 13 in., h = 16 in.

8. △MNP if m = 8 yd, n = 3.6 yd, p = 5.2 yd 9. △XYZ if x = 12 cm, y = 10 cm, z = 15.8 cm

Find the area of each triangle to the nearest tenth.

10. △RST if R = 115°, s = 15 yd, t = 20 yd 11. △MNP if n = 4 ft, P = 69°, N = 37°

12. △DEF if d = 2 ft, E = 85°, F = 19° 13. △JKL if j = 68 cm, l = 110 cm, K = 42.5°

N = 79°, m ≈ 9.4, p ≈ 13.6 R = 118°, r ≈ 74.7, s ≈ 46.1 E ≈ 75.0°, F ≈ 52.0°, d ≈ 40.5 A ≈ 40.5°, B ≈ 111.8°, C ≈ 27.7° about 25.9 ft 30 ft2 71.0 in2

7.2 yd2 60.1 cm2

135.9 yd2 11.9 ft2

0.7 ft2 2526.7 cm2

4-1 Study Guide and Intervention · Chapter 4 7 Glencoe Precalculus 4-1 Practice Right Triangle...

Documents

Transcript of 4-1 Study Guide and Intervention · Chapter 4 7 Glencoe Precalculus 4-1 Practice Right Triangle...