1. Simplify the expression so it is free of any trig or inverse trig

Transcription

1. Simplify the expression so it is free of any trig or inverse trig
1. Simplify the expression so it is free of any trig or inverse trig function. Use exact value.
a. sin−1 (0)
ans: 0
b. sin
ans:
−1
√ !
3
2
π
3
c. cos−1 (−1)
ans: π
d. cos−1
ans:
√ !
2
−
2
3π
4
e. tan−1 (−1)
π
ans: −
4
1
f. arcsin
2
π
ans:
6
√ !
2
g. arccos −
2
3π
4
2. Simplify the expression so it is free of any trig or inverse trig function. Use exact value.
ans:
a. arcsin(sin(3π))
ans: 0
b. arccos(cos(−
ans:
9π
))
2
π
2
c. tan−1 (tan(
ans: −
−21π
))
4
π
4
d. sin−1 (sin(4.579))
ans: −(4.579 − π) = π − 4.579
e. cos−1 (cos(−13))
ans: 13 − 4π
3. Simplify the expression so it is free of any trig or inverse trig function. Use exact value.
1
a. sin(arccos( ))
3
√
√
8
2 2
ans:
=
3
3
2
b. cos(arcsin(− ))
5
√
21
ans:
5
c. sec(tan−1 (3))
√
ans: 10
2
d. tan(cos−1 ( ))
7
√
√
45
3 5
ans:
=
2
2
3
e. csc(sin−1 ( √ ))
13
√
13
ans:
3
4. Rewrite the following in terms of x so that the expression is free of any trig or inverse trig function:
a. sin(arccos x)
p
ans: 1 − x2
b. cos(tan−1 x)
ans: √
1
1 + x2
c. tan(sin−1 x)
x
ans: √
1 − x2
5. Solve the given equation. If the range for x is indicated, only solve for x over that interval. Otherwise, solve
for all possible solutions.
√
2
a. cos x =
2
π
Ans: x = ± + 2πn
4
√
3
b. sin x = −
2
4π
5π
Ans: x =
+ 2πn or
+ 2πn
3
3
1
c. sin x = −
2
7π
11π
Ans: x =
+ 2πn or
+ 2πn
6
6
√
d. tan x = 3
π
Ans: x = + πn
3
e. cos x = −1
Ans: x = π + 2πn
f. sin x = 0
Ans: x = πn
g. 2 sin x + 1 = 0, 0 ≤ x ≤ 2π
11π
7π
or x =
6
6
√
h. 2 cos x − 3 = 0, 0 ≤ x ≤ 2π
Ans: x =
π
11π
or x =
6
6
i. cos x = cot x, 0 ≤ x ≤ 2π
Ans: x =
π
3π
or x =
2
2
j. tan x = −2 sin x, 0 ≤ x ≤ 2π
Ans: x =
Ans: x = 0 or x =
or x = π or x =
2π
3
4π
3
or x = 2π
k. 2 cos2 x + 3 sin x = 0, 0 ≤ x ≤ 2π
7π
11π
or x =
6
6
l. cos 2x = sin x
π
5π
3π
Ans: x = + 2πn or x =
+ 2πn or x =
+ 2πn
6
6
2
m. sin 2x = 1 + cos 2x
π
Ans: x = + πn
2
n. cos 2x = cos x − 1
π
π
Ans: x = ± + 2πn or x = + πn
3
2
Ans: x =

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