Answers and Explanations 165
Answers
101–200
106.
−4 7
7
To find the value of cscθ , you can find the value of sinθ and then use
csc
sin
θ
θ
= 1 . Use the identity sin
cos
2
2
1
θ
θ
+
= to solve for sinθ :
sin
sin
sin
2
2
2
2
3
4
1
1 9
16
7
16
θ
θ
θ
+ ( ) =
= −
=
Next, take the square root of both sides, keeping the negative solution for sine
because 3
2
2
π θ
π
< < :
sin
sin
2
7
16
7
4
θ
θ
=
= −
Therefore, using csc
sin
θ
θ
= 1 , you have
cscθ =
−
= − =
−
1
7
4
4
7
4 7
7
107.
− 80
89
To find the value of sin( )
2θ , you can use the identity sin( )
sin cos
2
2
θ
θ
θ
=
. Notice that
because sinθ > 0 and cosθ < 0, angle θ must be in the second quadrant. Using
tanθ = − 8
5
, you can make a right triangle and find the missing side using the
Pythagorean theorem. When making the triangle, you can neglect the negative sign:
h =
+
=
8 5
89
2
2