Principal Values
Find the exact value of each expression in radians.
__________ 1.
arc sin 2
1
__________ 11.
cot −1 1
__________ 2.
Arc sin 2
1
__________ 12.
Cot −1 1
__________ 13.
sec −1 √2
__________ 3.
1
arc cos 2
__________ 14.
Sec −1 √2
__________ 15.
arc cos (−
√2
)
2
Arc cos (−
√2
)
2
__________ 4.
__________ 5.
Arc cos
1
2
sin−1 0
__________ 6.
Sin
0
__________ 16.
__________ 7.
tan−1 (−√3)
__________ 17.
cos−1 (− 2)
__________ 8.
Tan−1(−√3)
__________ 18.
Cos −1 (− )
__________ 9.
arc sin (−
√3
)
2
__________ 19.
arc csc(−2)
__________ 10.
Arc sin (−
√3
)
2
__________ 20.
Arc csc(−2)
−1
1
1
2
Find the exact value of each expression.
__________ 21.
sin (cos −1
√2
)
2
__________ 27.
cos (arcsin
√3
)
2
__________ 22.
sin (Cos −1
√2
)
2
__________ 28.
cos (Arcsin
√3
)
2
__________ 23.
arcsin (sin
2𝜋
)
3
__________ 29.
sin (cos −1 (− ))
__________ 24.
Arcsin (sin
2𝜋
)
3
__________ 30.
sin (Cos −1 (− ))
__________ 25.
cos−1 (cos ( ))
4𝜋
3
__________ 31.
cos−1 (cos (− ))
__________ 26.
Cos −1 (cos ( 3 ))
__________ 32.
Cos −1 (cos (− 3 ))
4𝜋
1
2
1
2
𝜋
3
𝜋
Question
1
2
33.
sin−1
34.
tan−1 √3
35.
arccot 1
36.
cos−1
37.
arcsec 2
38.
csc −1 √2
√2
2
0 ≤ 𝑥 < 2𝜋
All Values
Principal Values
Principal Values KEY
Find the exact value of each expression in radians.
𝜋 5𝜋
,
6 6
__________
1.
𝜋
6
__________
2.
𝜋 5𝜋
,
3 3
__________
3.
𝜋
3
__________
4.
0, 𝜋
__________
5.
0
__________
6.
2𝜋 5𝜋
,
3 3 7.
__________
𝜋
−
3
__________
8.
4𝜋 5𝜋
,
3 3 9.
__________
𝜋
−
3
__________
10.
𝜋 5𝜋
,
4 4
__________
11.
1
arc sin 2
𝜋
__________
12.
4
𝜋 7𝜋
,
4 4
__________
13.
1
Arc sin 2
1
arc cos 2
1
Arc cos 2
sin−1 0
−1
Sin
0
−1
tan (−√3)
−1
Tan (−√3)
√3
arc sin (− 2 )
Arc sin (−
√3
)
2
cot −1 1
Cot −1 1
sec −1 √2
𝜋
4
__________
14.
Sec −1 √2
3𝜋 5𝜋
,
__________
4 4 15.
arc cos (−
√2
)
2
3𝜋
4
__________
16.
Arc cos (−
√2
)
2
2𝜋 4𝜋
,
3 3
__________
17.
cos−1 (− 2)
2𝜋
3
__________
18.
Cos −1 (− )
7𝜋 11𝜋
,
6 6 19.
__________
𝜋
−
6
__________
20.
1
1
2
arc csc(−2)
Arc csc(−2)
Find the exact value of each expression.
√2
±
2
__________
21.
√2
2
__________
22.
𝜋 2𝜋
,
3 3
__________
23.
𝜋
3
__________
24.
2𝜋 4𝜋
,
3 3
__________
25.
2𝜋
__________
26.
3
√2
sin (cos −1 2 )
√2
sin (Cos −1 2 )
2𝜋
arcsin (sin 3 )
2𝜋
Arcsin (sin )
3
cos
−1
4𝜋
(cos ( 3 ))
−1
4𝜋
(cos ( 3 ))
Cos
1
±
2
__________
27.
cos (arcsin
√3
)
2
1
__________
28.
2
cos (Arcsin
√3
)
2
√3
±
2
__________
29.
sin (cos −1 (− ))
√3
2
__________
30.
sin (Cos −1 (− 2))
𝜋 5𝜋
,
3 3
__________
31.
cos−1 (cos (− 3 ))
𝜋
3
__________
32.
Cos −1 (cos (− 3 ))
1
2
1
𝜋
𝜋
Question
0 ≤ 𝑥 < 2𝜋
All Values
Principal Values
33.
sin−1 2
𝜋 5𝜋
,
6 6
𝜋
5𝜋
± 2𝜋𝑛,
± 2𝜋𝑛
6
6
𝜋
6
34.
tan−1 √3
𝜋 4𝜋
,
3 3
𝜋
± 𝜋𝑛
3
𝜋
3
35.
arccot 1
𝜋 5𝜋
,
4 4
𝜋
± 𝜋𝑛
4
𝜋
4
36.
cos−1
√2
2
𝜋 7𝜋
,
4 4
𝜋
7𝜋
± 2𝜋𝑛,
± 2𝜋𝑛
4
4
𝜋
4
37.
arcsec 2
𝜋 5𝜋
,
3 3
𝜋
5𝜋
± 2𝜋𝑛,
± 2𝜋𝑛
3
3
𝜋
3
38.
csc −1 √2
𝜋 3𝜋
,
4 4
𝜋
3𝜋
± 2𝜋𝑛,
± 2𝜋𝑛
4
4
𝜋
4
1