Make the trigonometric substitution x = a csc θ for 0 < θ < π/2 and a > 0. Simplify the resulting expression.
Question # 00254010
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Updated on: 04/18/2016 08:42 AM Due on: 05/18/2016
Question 1
Question 3
4Make the trigonometric substitution
x = a csc θ for 0 < θ < π/2 and a > 0.
Simplify the resulting expression.
x2 − a2
x
5Make the trigonometric substitution
x = d cot θ for 0 < θ < π and d > 0.
Use fundamental identities to simplify the resulting expression.
d2 + x2
6Make the trigonometric substitution
x = a csc θ for 0 < θ < π/2 and a > 0.
Use fundamental identities to simplify the resulting expression.
x2 − a2
Question 7
Question 8
9Find the exact values of sin 2θ, cos 2θ, and tan 2θ for the given value of θ.
cos θ =
3
5
;
0° < θ < 90°
sin 2θ =
cos 2θ =
tan 2θ =
10If tan α = 5 and α is acute, find the exact value of sin 2α.
11Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.
sec θ = 5/4;
0° < θ < 90°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
12Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given condition.
csc θ = − 5/3;
−90° < θ < 0°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
13Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.
sec θ = − 4/3;
180° < θ < 270°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
14Use halfangle formulas to find the exact values.
(a)
cos 22°30'
(b)
sin 75°
(c)
tan
π
12
15Use halfangle formulas to find the exact values.
(a)
cos 165°
(b)
sin 112°30'
(c)
tan
π
8
16Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
sin−1
−
2
2
(b)
cos−1
−
1
2
(c)
tan−1
−
3
16Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
sin−1
−
2
2
(b)
cos−1
−
1
2
(c)
tan−1
−
3
17Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
arcsin
3
2
(b)
arccos
2
2
(c)
arctan
3
3
18Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
arcsin 0
(b)
arccos(−1)
(c)
arctan 0
Question 19
Question 20
Question 3
4Make the trigonometric substitution
x = a csc θ for 0 < θ < π/2 and a > 0.
Simplify the resulting expression.
x2 − a2
x
5Make the trigonometric substitution
x = d cot θ for 0 < θ < π and d > 0.
Use fundamental identities to simplify the resulting expression.
d2 + x2
6Make the trigonometric substitution
x = a csc θ for 0 < θ < π/2 and a > 0.
Use fundamental identities to simplify the resulting expression.
x2 − a2
Question 7
Question 8
9Find the exact values of sin 2θ, cos 2θ, and tan 2θ for the given value of θ.
cos θ =
3
5
;
0° < θ < 90°
sin 2θ =
cos 2θ =
tan 2θ =
10If tan α = 5 and α is acute, find the exact value of sin 2α.
11Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.
sec θ = 5/4;
0° < θ < 90°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
12Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given condition.
csc θ = − 5/3;
−90° < θ < 0°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
13Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.
sec θ = − 4/3;
180° < θ < 270°
sin(θ/2) =
cos(θ/2) =
tan(θ/2) =
14Use halfangle formulas to find the exact values.
(a)
cos 22°30'
(b)
sin 75°
(c)
tan
π
12
15Use halfangle formulas to find the exact values.
(a)
cos 165°
(b)
sin 112°30'
(c)
tan
π
8
16Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
sin−1
−
2
2
(b)
cos−1
−
1
2
(c)
tan−1
−
3
16Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
sin−1
−
2
2
(b)
cos−1
−
1
2
(c)
tan−1
−
3
17Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
arcsin
3
2
(b)
arccos
2
2
(c)
arctan
3
3
18Find the exact value of the expression whenever it is defined. (If an answer is undefined,
enter UNDEFINED.)
(a)
arcsin 0
(b)
arccos(−1)
(c)
arctan 0
Question 19
Question 20

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Solution: Make the trigonometric substitution x = a csc θ for 0 < θ < π/2 and a > 0. Simplify the resulting expression.