MAT1275-Spring 2023
Practice Exam 3
City Tech - CUNY
Problem 1. (a). Convert −150◦ into radian measure.
radians into degree.
(b). Convert 3π
4
(c). Convert 390◦ degree into radian measure.
−150◦ = −150 ×
π
5π
=−
180
6
3π
3 × 180◦
=
= 135◦
4
4
390◦ = 390 ×
π
13π
=
180
6
Problem 2. An airplane is flying 700 feet above ground. From the plane to the base of the
control tower, the angle of depression is 47◦ . How far away is ground directly underneath
the plane to the control tower? Round your answer to the nearest tenth.
∠A = 90◦ − 47◦ = 43◦ ,
tan 43◦ =
|AB| = 700 f eet
Opp
|BC|
|BC|
=
=
,
Adj
700
|AB|
|BC| = tan 43◦ · 700 ≈ 652.8 f eet
Problem 3. Consider the following right triangle. If AB has length 4, find:
(a). the length of AC,
(b). the length of BC.
Practice Exam 3
MAT1275-Spring 2023
tan ∠C = tan 45◦ =
Opp
|AB|
=
,
Adj
|AC|
1=
Page 2 of 3
4
,
|AC|
|AC| = 4
√
2
4
=
,
2
|BC|
Opp
|AB|
sin ∠C = sin 45 =
=
,
hyp
|BC|
◦
|BC| =
4
√
2
2
√
2
=4× √ =4 2
2
Problem 4. Find the exact solutions to the trig equations for x ∈ [0, 2π):
√
(a). 6 cos x − 3 = 0,
(b). 6 tan x = −2 3.
(a). 6 cos x − 3 = 0,
√
(b). 6 tan x = −2 3,
√
tan x = −
Problem 5. Prove the identities:
1
1
−
= tan x · sin x,
(a).
cos x sec x
(a).
1
cos x = ,
2
6 cos x = 3,
3
,
3
x=
5π
,
6
x=
π
,
3
5π
3
11π
6
(b). csc x − sin x = cot x · cos x,
(c). sec x · csc x = tan x + cot x
1
1
1
1
cos2 x
1 − cos2 x
sin2 x
sin x
−
=
− cos x =
−
=
=
=
· sin x = tan x · sin x
cos x sec x
cos x
cos x
cos x
cos x
cos x
cos x
(b). csc x − sin x =
1
1
sin2 x
1 − sin2 x
cos2 x
sin x
− sin x =
−
=
=
=
· cos x = cot x · cos x
sin x
sin x
sin x
sin x
cos x
cos x
1
1
1
sin2 x + cos2 x
(c). sec x · csc x =
·
=
=
cos x sin x
sin x · cos x
sin x · cos x
sin2 x
cos2 x
sin x cos x
=
+
=
+
= tan x + cot x
sin x · cos x sin x · cos x
cos x sin x
Cont.
Practice Exam 3
MAT1275-Spring 2023
Page 3 of 3
Problem 6. For the given expressions
(1). Identify quadrant in which angle is located.
(2). Find reference angle.
(3). Calculate the exact value.
5π
(b). tan −
4
◦
(a). cos 330 ,
(a). 330◦ is in the quadrant 4, and the reference angle is
√
3
0
◦
◦
◦
◦
◦
t = 360 − 330 = 30 ,
cos 330 = cos 30 =
.
2
(b). − 5π
is in the quadrant 2, and the reference angle is
4
π 5π
π
5π
0
t =
−π = ,
tan −
= cos 30◦ = − tan
= −1.
4
4
4
4
Problem 7. For the information given, state the value of the five remaining trig functions
of θ :
2
(a) cos θ = and tan θ < 0,
(b) tan θ = 2 and sin θ < 0
3
2
(a) cos θ = ,
r
√3
5
5
sin θ = ±
=±
,
9
3
cos2 θ + sin2 θ = 1,
since
tan θ < 0,
√
√
√
− 35
5 3
5
sin θ
tan θ =
= 2 =−
· =−
,
cos θ
3
2
2
3
sec θ =
1
1
3
= 2 = ,
cos θ
2
3
csc θ =
4
+ sin2 θ = 1,
9
θ is in the quadrant 4,
cot θ =
4
5
= ,
9 √9
5
sin θ = −
3
√
2 5
=−
5
sin2 θ = 1 −
1
2
= −√
tan θ
5
1
1
3
= √ = −√
sin θ
5
− 35
√
√
sec2 t = 5, sec t = ± 5, sec t = − 5
√
√
√
1
1
5
5
2 5
cos t =
= −√ = −
, sin t = tan t · cos t = 2 · (−
)=−
,
sec t
5
5
5
5
√
5
1
1
− 5
1
1
√ = √ =
csc t =
=
, cot t =
=
sin t
2
tan t
2
2 5
−255
(b). 1 + 22 = sec2 t,
Problem 8. Given 4ABC, answer the following (round each answer to the nearest tenth):
(a). If ∠A = 50◦ , ∠B = 75◦ , and a = 20, find side b.
(b). If b = 9, c = 6, and ∠A = 67, find side a
.
The End.