Precalc 1720

Find the exact values cos 7𝜋/12

1/4(sqrt2 - sqrt6)

Find the supplementary angle to: 85°13'54" 82.3

94°46'6"

csc 𝜃 > 0 and sec 𝜃 > 0 a)quadrant | b) quadrant || c)quadrant ||| d) quadrant ||||

a

Find the amplitude and period y = sin 8x

amp = 1 period = 𝜋/4

Find all solutions of the equation. 3 sin x = 3𝜋/2

NO SOLUTION

Find all solutions of the equation. tan 𝜃 = sqrt3

𝜃 = 𝜋/3 + 𝜋n for n = 0, ±1, ±2,

Find the solutions of the equation that are in the interval [0, 2𝜋). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 5 − 5 sin t = 5sqrt3 cos t

𝜋/2 , 11𝜋/6

Find the exact degree measure of: 𝜋/3 11𝜋/6 𝜋/4

𝜋/3 : 60° 11𝜋/6 : 330° 𝜋/4 : 45°

Use an addition or subtraction formula to find the solutions of the equation that are in the interval [0, 𝜋). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) tan t − tan 4t = 1 + tan 4t tan t

𝜋/4, 7𝜋/12, 11𝜋/12

Refer to the graph of y = sin x or y = cos x to find the exact values of x in the interval [0, 4𝜋] that satisfy the equation. (Enter your answers as a comma-separated list.) 9 sin x = 9/2

𝜋/6 , 5𝜋/6 , 13𝜋/6 , 17𝜋/6

Find the solutions of the equation that are in the interval [0, 2𝜋). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 2 sin^2 u = −1 + 3 sin u

𝜋/6 , 𝜋/2 , 5𝜋/6

Find the exact values tan 𝜋/4 − tan 5𝜋/6

1 + 1/sqrt3

Find the exact values. cos 𝜋/3 + cos 𝜋/4

1/2(1 + sqrt2)

Find the exact values sin 11𝜋/12

1/4(sqrt6 - sqrt2)

Find two positive and two negative coterminal angles for the following angles: 110° 145° -50°

110: 470°, 830° ; -250°, -610° 145 :505°, 865° ; -215°, -575° -50 : 310°, 670°; -410°, -770°

Find the exact radian measure of the angles: 150° -20° 195°

150 : 5𝜋/6 -20 : -𝜋/9 195 : 13𝜋/12

Find the angle complementary to the following angles: 72°7'51" 5.02

17°52'9'' 84.98

Find the exact value tan −7𝜋/12

2 + sqrt3

From a point 35 meters above level ground, a surveyor measures the angle of depression of an object on the ground at 60°. Approximate the distance from the object to the point on the ground directly beneath the surveyor. (Round your answer to two decimal places.)

20.21 m

Express terms in degrees, minutes, and seconds 3.3 4.7

3.3 : 189°4'34'' 4.7 : 269°17'25"

A pilot, flying at an altitude of 7000 feet, wishes to approach the numbers on a runway at an angle of 13°. Approximate, to the nearest 100 feet, the distance from the airplane to the numbers at the beginning of the descent.

31,100

A rocket is fired at sea level and climbs at a constant angle of 80° through a distance of 10,000 feet. Approximate its altitude to the nearest foot.

9,848

Use half-angle formulas to find the exact values. a) cos 67°30' b) sin 75° c) tan 𝜋/12 d) cos 105° e) tan 5𝜋/12

a) (sqrt2-sqrt2)/2 b) (sqrt2+sqrt3)/2 c) 2-sqrt3 d) -1/2(sqrt2-sqrt3) e) 2 + sqrt3

Let P(t)be the point on the unit circle U that corresponds to t. If P(t)has the given rectangular coordinates 3/5, 4/5, find: (a) P(t + 𝜋) (b) P(t − 𝜋) (c) P(−t) (d) P(−t − 𝜋)

a) -3/5 , -4/5 b) -3/5 , -4/5 c) 3/5 , -4/5 d) -3/5 , 4/5

Approximate to four decimal places, when appropriate. a) cot(𝜋/15) b) csc 2.04 c) cos(-9.02) d) tan(3𝜋/8)

a) 4.7046 b) 1.1212 c) -.9192 d)2.4142

Find the reference angle 𝜃R if 𝜃 has the given measure. a) 3𝜋/4 b) 5𝜋/3 c) −𝜋/6 d) 9𝜋/4

a) 𝜋/4 b) 𝜋/3 c)𝜋/6 d) 𝜋/4

Find the amplitude and period y = 2 sin 1/8x

amp = 2 period = 16𝜋

Find the amplitude, period, and phase shift y = 5 sin(2x - 𝜋/3)

amp = 5 period = 𝜋 phase shift = 𝜋/6

cos 𝜃 < 0 and sin 𝜃 > 0 a)quadrant | b) quadrant || c)quadrant ||| d) quadrant ||||

b

sec 𝜃 < 0 and tan 𝜃 < 0 a)quadrant | b) quadrant || c)quadrant ||| d) quadrant ||||

b

Express as a trigonometric function of one angle. cos 56° cos 23° + sin 56° sin 23°

cos 65°

sin = 24/25 find the remaining trigonometric functions

cos = 7/25 tan = 24/7 csc = 25/24 sec = 25/7 cot = 7/24

Find the exact values of sin 2𝜃, cos 2𝜃, and tan 2𝜃 for the given value of 𝜃. cos 𝜃 = 3/5

sin 2𝜃 = 24/25 cos 2𝜃 = -7/25 tan 2𝜃 = -24/7

Find the exact values of the six trigonometric functions of 𝜃 if 𝜃 is in standard position and the terminal side of 𝜃 is in the given quadrant and satisfies the given condition. bisects quadrant ||

sin = 1/sqrt2 cos = -1/sqrt2 tan = -1 csc = sqrt2 sec = -sqrt2 cot = -1

A point P(x, y) is shown on the unit circle U corresponding to a real number t. Find the values of the trigonometric functions at t. a = -3/5 , b = 4/5

sin = 4/5 cos = -3/5 tan = -4/3 csc = 5/4 sec = -5/3 cot = -3/4

Use fundamental identities to find the values of the trigonometric functions for the given conditions. tan 𝜃 = − 4/3and sin 𝜃 > 0

sin = 4/5 cos = -3/5 tan = -4/3 csc = 5/4 sec = -5/3 cot = -3/4

Find the values of the six trigonometric functions. a = 1 , c = 9

sin = 4sqrt5/9 cos = 1/9 tan = sqrt80 csc = 9/4sqrt5 sec = 9 cot = 1/4sqrt5

Find the exact values of sin(𝜃/2), cos(𝜃/2), and tan(𝜃/2) for the given conditions. sec 𝜃 = 5/3; 0° < 𝜃 < 90°

sin(𝜃/2) = sqrt1/5 cos(𝜃/2) = 2/sqrt5 tan(𝜃/2) = 1/2

Find all solutions of the equation sin x = sqrt2/2

x = 𝜋/4 + 2𝜋n, and x = 3𝜋/4 + 2𝜋n for n = 0, ±1, ±2,...

Write the equation for: amplitude = 2 period = 2𝜋 phase shift = -𝜋

y = 2 sin(1x + 𝜋)

Given the indicated parts of triangle ABC with 𝛾 = 90°, approximate the remaining parts. a = 27 , b = 58 𝛼 = 𝛽 = c =

𝛼 = 25° 𝛽 = 65° c = 64

Find all solutions of the equation. 2 cot 𝛼 = − 2/sqrt3

𝛼 = 2𝜋/3 + 𝜋n for n = 0, ±1, ±2

Given the indicated parts of triangle ABC with 𝛾 = 90°, find the exact values of the remaining parts. 𝛽 = 45°, b = 40 𝛼 = a = c =

𝛼 = 45° a = 40 c = 40sqrt2

Find the solutions of the equation that are in the interval [0, 2𝜋). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) cot2 𝜃 − sqrt3 cot 𝜃 = 0

𝜋/6 , 𝜋/2 , 7𝜋/6 , 3𝜋/2

Express as a trigonometric function of one angle. cos 7 sin(−8) − cos 8 sin 7

-sin 15

Find the exact values sin 2𝜋/3 + sin 𝜋/4

1/2(sqrt2 + sqrt3)

Approximate, to the nearest 0.1°, all angles 𝜃 in the interval [0°, 360°) that satisfy the equation. (Enter your answers as a comma-separated list.) a) sin 𝜃 = −0.2550 b) cos 𝜃 = 0.9580 c) tan 𝜃 = 2.775 d) cot 𝜃 = −0.9704 e) sec 𝜃 = −1.111 f) csc 𝜃 = 1.388

a) 194.8° , 345.2° b) 16.7° , 343.3° c)70.2° , 250.2° d) 134.1° , 314.1° e) 154.2° , 205.8° f) 46.1° , 133.9°

Find the reference angle 𝜃R if 𝜃 has the given measure. a) 290° b) 210° c)-238° d) -650°

a) 70° b) 30° c) 58° d) 70°

Find the reference angle 𝜃R if 𝜃 has the given measure a) 5 b) −2 c) 2.5 d) 58

a) 73.5° b)65.4° c) 36.8° d) 83.2°

Express as a cofunction of a complementary angle. a) cos 5𝜋/8 b) sin 1/2 c) tan 5 d) csc .62

a) sin -𝜋/8 b) cos (𝜋-1)/2 c) cot (𝜋-10)/2 d) sec (𝜋/2 - .62)

f(x) = cos 6x + sqrt3 sin 6x express in terms of the cosine function and identify amplitude, period, and phase shift

f(x) = 2cos (6x-𝜋/3) amp = 2 period = 𝜋/3 phase shift = 𝜋/18

Find the trigonometric functions t = −5𝜋

sin = 0 cos = -1 tan = 0 csc = UNDEFINED sec = -1 cot = UNDEFINED

Given the indicated parts of triangle ABC with 𝛾 = 90°, find the exact values of the remaining parts. a = 5sqrt3, c = 10 𝛼 = 𝛽 = b =

𝛼 = 60° 𝛽 = 30° b = 5

Find all solutions of the equation. 4 sec2 𝛼 − 16 = 0

𝛼 = 𝜋/3 + 𝜋n, 2𝜋/3 + 𝜋n for n = 0, ±1, ±2

Find all solutions of the equation. (2 sin 𝜃 + 1)(3 cos 𝜃 + 6) = 0

𝜃 = 7𝜋/6 + 2𝜋n, 11𝜋/6 + 2𝜋n for n = 0, ±1, ±2

Shown in the figure is the screen for a simple video arcade game in which ducks move from A to B at the rate of 7 cm/sec. Bullets fired from point O travel 30 cm/sec. If a player shoots as soon as a duck appears at A, at which angle φ should the gun be aimed in order to score a direct hit? (Round your answer to one decimal place.)

13.5°

Refer to the graph of y = sin x or y = cos x to find the exact values of x in the interval [0, 4𝜋] that satisfy the equation. (Enter your answers as a comma-separated list.) 8 sin x = −8

3𝜋/2 , 7𝜋/2

Find two positive angles and two coterminal angles: 3𝜋/4 -7𝜋/5

3𝜋/4: 11𝜋/4, 19𝜋/4 ; -5𝜋/4, -13𝜋/4 -7𝜋/5: 3𝜋/5, 13𝜋/5 ; -17𝜋/5, -27𝜋/5

angle = 45 , r = 11 a) find the length of the arc b) find the area of the sector

a) 8.64 b)47.52

If sin 𝛼 = − 35 and sec 𝛽 = 54 for a third-quadrant angle 𝛼 and a first-quadrant angle 𝛽, find the following. a) sin(𝛼 + 𝛽) b) cos(𝛼 + 𝛽) c) the quadrant containing 𝛼 + 𝛽

a) sin(𝛼 + 𝛽) = -24/25 b) cos(𝛼 + 𝛽) = 24/7 c) the quadrant containing 𝛼 + 𝛽 = Quadrant 3

If 𝛼 and 𝛽 are acute angles such that cos 𝛼 = 15/17 and tan 𝛽 = 3/4, find the following. a) sin(𝛼 + 𝛽) b) cos(𝛼 + 𝛽) c) the quadrant containing 𝛼 + 𝛽

a) sin(𝛼 + 𝛽) = 77/85 b) cos(𝛼 + 𝛽) = 36/85 c) the quadrant containing 𝛼 + 𝛽 = Quadrant |

Find all solutions of the equation. cos 𝜃 = 1/sec 𝜃

all 𝜃 except 𝜃 = 𝜋/2 + 𝜋n for n = 0, ±1, ±2,...

Find the amplitude, period, and phase shift y = sin( x + 𝜋/3)

amp = 1 period = 2𝜋 phase shift = -𝜋/3

Find the amplitude, period, and phase shift y = -2 cos(2x + 𝜋/3)

amp = 2 period = 𝜋 phase shift = -𝜋/6

Find the amplitude and period y = 5 sin x

amp = 5 period = 2𝜋

Express in terms of the cosine function with exponent 1. cos^4 𝜃/2

cos(𝜃)/2 + 1/8cos(2𝜃) + 3/8

sin 𝜃 < 0 and cot 𝜃 < 0 a)quadrant | b) quadrant || c)quadrant ||| d) quadrant ||||

d

If a circular arc of the given length s subtends the central angle on a circle, find the radius of the circle s = 9cm , angle = 6

radius = 1.5cm

Express as a trigonometric function of one angle. sin 53° cos 8° + cos 53° sin 8°

sin 61°

find the six trigonometric functions of the angle with the standard coordinates 5,-12

sin = -12/13 cos = 5/13 tan = -12/15 csc = -13/12 sec = 13/5 cot = -5/12

Find the exact values of the six trigonometric functions of 𝜃 if 𝜃 is in standard position and the terminal side of 𝜃 is in the specified quadrant and satisfies the given condition. |||; parallel to the line 3y-5x+9=0

sin = -5/sqrt34 cos = -3/sqrt34 tan = 5/3 csc = -sqrt34/5 sec = -sqrt34/3 cot = 3/5

Find the trigonometric functions t = 4𝜋

sin = 0 cos = 1 tan = 0 csc = UNDEFINED sec = 1 cot = UNDEFINED

Use a formula for negatives to find the exact value sin(-270) cos(-3𝜋/4) tan(-45) sin(-3𝜋/2)

sin = 1 cos = -1/sqrt2 tan = -1 sin = 1

Find the exact values of sin(𝜃/2), cos(𝜃/2), and tan(𝜃/2) for the given condition. 3 tan 𝜃 = 3; −180° < 𝜃 < −90°

sin(𝜃/2)= -1/2sqrt2+sqrt2 cos(𝜃/2)= 1/2sqrt2-sqrt2 tan(𝜃/2)= -sqrt2 - 1

Find all solutions of the equation. 2 cos t = −2

t = (𝜋 + 2𝜋n) for n = 0, ±1, ±2,...

Find all solutions of the equation −4 cos t − 2 = 0

t = 2𝜋/3 + 2𝜋n, 4𝜋/3 + 2𝜋n for n = 0, ±1, ±2,

Complete the statement by referring to a graph of a trigonometric function. As x → −7𝜋, tan x → As x → (9𝜋/2)+, tan x → As x → (𝜋/2)−, sec x → As x → 0, sec x →

0 -infinity infinity 1

Use an addition or subtraction formula to find the solutions of the equation that are in the interval [0, 𝜋). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) sin 5t cos 2t = sin 2t cos 5t

0, 𝜋/3, 2𝜋/3