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

Question 1

Question 3

4-Make the trigonometric substitution
x = a csc θ for 0 &lt; θ &lt; π/2 and a &gt; 0.
Simplify the resulting expression.
x2 − a2
x

5-Make the trigonometric substitution
x = d cot θ for 0 &lt; θ &lt; π and d &gt; 0.
Use fundamental identities to simplify the resulting expression.
d2 + x2

6-Make the trigonometric substitution
x = a csc θ for 0 &lt; θ &lt; π/2 and a &gt; 0.
Use fundamental identities to simplify the resulting expression.
x2 − a2

Question 7

Question 8

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

0° &lt; θ &lt; 90°

sin 2θ =

cos 2θ =
tan 2θ =
10-If tan α = 5 and α is acute, find the exact value of sin 2α.

11-Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.
sec θ = 5/4;
0° &lt; θ &lt; 90°

sin(θ/2) =

cos(θ/2) =

tan(θ/2) =
12-Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given condition.
csc θ = − 5/3;

−90° &lt; θ &lt; 0°

sin(θ/2) =

cos(θ/2) =

tan(θ/2) =

13-Find the exact values of sin(θ/2), cos(θ/2), and tan(θ/2) for the given
conditions.

sec θ = − 4/3;

180° &lt; θ &lt; 270°

sin(θ/2) =

cos(θ/2) =

tan(θ/2) =

14-Use half-angle formulas to find the exact values.
(a)
cos 22°30'

(b)
sin 75°

(c)
tan

π
12

15-Use half-angle formulas to find the exact values.
(a)
cos 165°

(b)
sin 112°30'

(c)
tan

π
8

16-Find 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

16-Find 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

17-Find 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

18-Find 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