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Math 102: trigonometry
Instructor: Julio C. Herrera
Exam 3
Name:
May 27, 2016
Exam Score:
Instructions: This exam covers the material from chapter five and six. Please read each
question carefully before you attempt to solve them. Remember that you have to show all
of your work clearly in order to get credit. The exam is closed book and calculators are not
allowed. Good luck!
Problem 1: Verify the following identity.
(cos x + cos y)2 + (sin x − sin y)2 = 2 + 2 cos(x + y)
Solutions:
Recall the sum-to-product identities:
x+y
x−y
cos x + cos y = 2 cos
cos
2
2
sin x − sin y = 2 cos
x+y
x−y
sin
2
2
With these formulas in mind, we will proceed starting from the left hand side of the identity
that we want to verify.
(cos x + cos y)2 + (sin x − sin y)2
2 2
x−y
x−y
x+y
x+y
cos
sin
+ 2 cos
= 2 cos
2
2
2
2
x+y
x−y
x+y
x−y
cos2
+ 4 cos2
sin2
2
2
2
2
x+y
x−y
x−y
= 4 cos2
cos2
+ sin2
2
2
2
= 4 cos2
= 4 cos2
"
x+y
(1)
2
1 + cos 2
=4
2
x+y
2
#
(half angle formula)
= 2 [1 + cos(x + y)]
= 2 + 2 cos(x + y)
Problem 2: Verify the identity using fundamental identities.
(csc x − cot x)(csc x + cot x)
= cos x
sec x
Page 1 of 6
Math 102: trigonometry
Instructor: Julio C. Herrera
Exam 3
May 27, 2016
Solutions:
We will start with the left hand side.
(csc x − cot x)(csc x + cot x)
sec x
csc2 x + csc x cot x − cot x csc x − cot2 x
=
sec x
=
(csc2 x − cot2 x) + csc x cot x − cot x csc x
sec x
=
1+0
sec x
1
sec x
= cos x
=
Problem 3: Determine the domain and range for the function f whose graph is given, and
use this information to state the domain and range of f −1 . Then sketch the line y = x and
sketch the graph of f −1 (x) on the same grid.
Solutions:
f : Domain [−3, 4] and Range [−3, 3]
f −1 : Domain [−3, 3] and Range [−3, 4]
The graph of f −1 is colored red.
Page 2 of 6
Math 102: trigonometry
Instructor: Julio C. Herrera
Exam 3
May 27, 2016
Problem 4: Evaluate the expression by drawing a right triangle and labeling the sides appropriately.
"
sin sec−1
!#
√
64 + x2
x
Solutions:
We essentially want to know what sin θ equals. Let’s work from the inside out.
!
√
2
64
+
x
sec−1
= θ implies that
x
!
√
64 + x2
sec θ =
x
hyp
we can create a triangle and solve for the sides of the
adj
triangle using the Pythagorean theorem.
Since we know that sec θ =
Page 3 of 6
Math 102: trigonometry
Instructor: Julio C. Herrera
Exam 3
May 27, 2016
Now that we know all of the sides of the triangle we just need to find sin θ.
8
opp
=√
.
sin θ =
hyp
64 + x2
Hence,
!#
√
8
64 + x2
=√
x
64 + x2
"
sin sec−1
Problem 5: Surfing the Perfect Wave: For a wave to be surfable, it can’t break all at
once. Robert Guza and Tony Bowen have shown that a wave has a surfable shoulder if it
hits the shore-line at an angle θ given by
−1
θ = sin
1
(2n + 1) tan β
where β is the angle at which the beach slopes down and where n = 0, 1, 2, ...
a. For β = 10◦ , find θ when n = 3.
b. For β = 15◦ , find θ when n = 2, 3, and 4.
c. Explain why the formula does not give a value for θ when n = 0 or 1.
Solutions:
−1
a. θ = sin
1
7 tan 10◦
= sin−1 (.79) ≈ 52.53◦
b. For β = 15◦ , find θ when n = 2, 3, and 4:
n = 2 : θ = 48.62◦
n = 3 : θ = 32◦
n = 4 : θ = 24.20◦
Page 4 of 6
Math 102: trigonometry
Instructor: Julio C. Herrera
Exam 3
May 27, 2016
c. For n = 0 we get θ = 3.73, which is impossible because the range of sin is [−1, 1].
The same reasoning applies for n = 1.
Problem 6: Verify the identity:
(sec θ − 1)(sec θ + 1) = tan2 θ
Solutions:
(sec θ − 1)(sec θ + 1)
= sec2 θ + sec θ − sec θ − 1
= sec2 θ − 1
= tan2 θ
Problem 7: Solve the following equation in [0, 2π) using the factor by grouping method.
Round nonstandard values to four decimal places.
√
√
4 3 sin2 x sec x − 3 sec x + 2 = 8 sin2 x
Solutions:
√
√
4 3 sin2 x sec x − 3 sec x + 2 = 8 sin2 x
√
√
⇒ 4 3 sin2 x sec x − 3 sec x + 2 − 8 sin2 x = 0
√
√
⇒ (4 3 sin2 x sec x − 3 sec x) + (2 − 8 sin2 x) = 0
√
⇒ 3 sec x(4 sin2 x − 1) + 2(1 − 4 sin2 x) = 0
√
⇒ 3 sec x(4 sin2 x − 1) − 2(4 sin2 x − 1) = 0
√
⇒ (4 sin2 x − 1)( 3 sec x − 2) = 0
√
⇒ 4 sin2 x − 1 = 0 or 3 sec x − 2 = 0
⇒ sin2 x =
⇒ sin x =
⇒x=
√
1
or 3 sec x = 2
4
1
2
or sec x = √
2
3
π 5π
π 11π
,
or x = ,
6 6
6 6
It might be helpful to graph sin x and sec x on [0, 2π).
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