Download SHORT ANSWER. Write the word or phrase that best completes

MATH 1316 Review for Test 3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use the fundamental identities to find the value of the trigonometric function.
2
1) Find sin if cos = and is in quadrant IV.
3
2) Find cos
if tan
= 6 and sin
< 0.
1)
2)
Complete the sentence so the result is an identity. Let x be any real number.
+ tan2 x = sec2 x
3)
3)
Perform the transformation.
4) Write sec x in terms of tan x.
4)
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
5) (1 + cot )(1 - cot ) - csc2
5)
Perform the indicated operations and simplify the result so there are no quotients.
1
6) sec sec
7)
sin
1 + sin
-
sin
1 - sin
6)
7)
Factor the trigonometric expression and simplify.
8) sec4 x - 2 sec2 x tan2 x + tan4 x
8)
Use the fundamental identities to simplify the expression.
cos2
+ csc sin
9)
sin2
9)
10) cos (-x) cos x - sin (-x) sin x
10)
Verify that each equation is an identity.
11) tan · csc = sec
11)
Use Identities to find the exact value.
12) cos 165°
12)
13) cos -
7
12
13)
Use identities to write each expression as a function of .
14) cos ( - )
14)
1
Find the exact value of the expression using the provided information.
1
1
15) Find cos(s + t) given that cos s = , with s in quadrant I, and sin t = - , with t in quadrant
3
2
15)
IV.
Write in terms of the cofunction of a complementary angle.
16) cos
16)
24
17) tan 80°
17)
Use the cofunction identities to find an angle
18) tan = cot (30°+ 5 )
that makes the statement true.
Verify that the equation is an identity.
3
1
cos x - sin x
=
19) cos x +
6
2
2
18)
19)
Use a sum or difference identity to find the exact value.
tan 80° + tan 70°
20)
1 - tan 80° tan 70°
20)
21) sin 15°
21)
Find the exact value of the expression using the provided information.
1
1
22) Find sin(s - t) given that cos s = , with s in quadrant I, and sin t = - , with t in
3
2
22)
quadrant IV.
23) Find tan(s - t) given that cos s = -
5
8
, with s in quadrant II, and sin t =
, with t in
13
17
23)
quadrant II.
Using a sum or difference identity, write the following as an expression involving functions of x.
24) sin (360° + x)
24)
Verify that the equation is an identity.
25) tan
2
25)
+ x = -cot x
Use an identity to write the expression as a single trigonometric function or as a single number.
26) 2 cos2 22.5° - 1
27) sin 8x cos 8x
26)
27)
2
Use identities to find the indicated value for each angle measure.
5
Find sin .
28) cos 2 = and terminates in quadrant I
6
Verify that each equation is an identity.
csc2
29) sec(2 )=
csc2 - 2
28)
29)
Write the product as a sum or difference of trigonometric functions.
30) cos 34° sin 25°
30)
Rewrite the following as a product of trigonometric functions.
31) sin 19° + sin 21°
31)
Find the exact value by using a half-angle identity.
32) sin 22.5°
32)
33) tan 75°
33)
Determine all solutions of the equation in radians.
x
1
34) Find cos , given that cos x = and x terminates in 0 < x < .
2
4
2
Use an identity to write the expression as a single trigonometric function or as a single number.
1 - cos 66°
35)
2
Verify that the equation is an identity.
x
x 2
= 1 - sin x
36) cos - sin
2
2
34)
35)
36)
Find the exact value of the real number y.
3
37) y = sin-1
2
37)
Give the degree measure of .
38) = csc-1 (2)
38)
Use a calculator to give the value to the nearest degree.
39) = sin-1 (0.2079)
39)
Use a calculator to give the real number value. Round the answer to 7 decimal places.
40) y = arcsec (2.8842912)
40)
3
Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
41) sin (cos-1 0.8324)
41)
Give the exact value of the expression.
12
42) csc sin-1
20
42)
Solve the equation for exact solutions over the interval [0, 2 ).
43) cos2 x + 2 cos x + 1 = 0
43)
Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
44) 4 sin2 = 3
44)
Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate answers
in radians to four decimal places and approximate answers in degrees to the nearest tenth.
45) 4 sin2 x - 1 = 0
45)
4
Answer Key
Testname: UNTITLED1
1) -
5
3
2) -
37
37
3) 1
tan2 x + 1
5) -2 cot2
4) ±
6) sin tan
7) -2 tan2
8) 1
9) csc2
10) 1
11) tan
· csc
12)
- 64
13)
24
=
sin
cos
·
1
sin
=
1
cos
= sec
2
6
14) -cos
3+2 2
15)
6
11
24
16) sin
17) cot 10°
18) = 10°
19) cos x +
20) -
6
64
22)
2 6+1
6
6
=
3
1
cos x - sin x.
2
2
140
171
25) tan
27)
- sin x sin
2
24) sin x
26)
6
3
3
21)
23) -
= cos x cos
2
+x =
sin (( /2) + x)
sin ( /2) cos x + sin x cos ( /2)
1 · cos x + sin x · 0
=
=
= -cot x.
cos (( /2) + x) cos ( /2) cos x - sin ( /2) sin x
0 · cos x - 1 · sin x
2
2
1
sin 16x
2
28) sin
=
3
6
5
Answer Key
Testname: UNTITLED1
29) sec(2 ) =
30)
1
1
=
cos(2 ) 1 - 2 sin2
=
1
sin2
1
sin2
-2
=
csc2
csc2 - 2
1
(sin 59° - sin 9°)
2
31) 2 sin 20° cos(-1°)
1
2- 2
32)
2
33) 2 + 3
10
34)
4
35) sin 33°
36) cos
1-2
37)
x
x 2
- sin
=
2
2
1 + cos x
2
1 - cos x 2 1 + cos x
1 + cos x
=
-2
2
2
2
sin2 x
sin x
=1-2
= 1 - sin x
4
2
3
38) 30°
39) 12°
40) 1.2167397
41) 0.5542
20
42)
12
43) }
44) {60°, 120°, 240°, 300°}
5
+n ,
+n
45)
6
6
6
1 - cos x 1 - cos x
1 - cos2 x
+
=1-2
=
2
2
4