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TRIGONOMETRY MODULE: PROBLEMS
1. Fill in the table given below by the appropriate measures:
Degrees
300o
150o

6
Radians
36o
3
4
7
6
2. The radius of Earth is about 6367 km. Calculate the length of the equator corresponding to the central angle of 1o
3. In the unit circle for the directed arcs in standard position given below, find the coordinates of the terminal points of each:
a)

3
b) −
7
4
c) −

4
d)
41 
6
e)
5
4
7
6
f)
g) −
5
6
h) −
2
3
4. Find the radian measure of the angle that corresponds to the coordiantes of the terminal points on the unit circle given below:
a) 
 3 , 1  b)  1 , 1  c)  −1 ,  3  d)  − 3 , 1  e)  −1 , 1  f)  −1 ,− 1  g)  1 ,−  3  h) 0 , 1
2 2
2 2
2 2
2
2
2 2
2 2
2 2
5. In the exercises below, determine the quadrant in which the angle
a) sin 0 and cos 0
b) sin 0 and cos 0
a) sin 150o cos 240 otan 210 ocot 300o
6. Evaluate

7. For the acute angle
8. Let ∈0,


2
and
4
5
. Evaluate
9. Evaluate the following trignometric ratios for
10. If
tan x=
3
4
d) cos 0 and tan 0
b) sin −300 o cos−240o tan −120o cot −210o .
∈

, 
2
and
cos =−
and x is an acute angle, then find the following:
12
:
13
a) sin x
a) sin 
b) tan x
b) tan 
c) cot x
P
T
Four equal squares
form the letter L as
shown in figure on
the left.
Find tan (NTL)
N
L
d) sec x
K
M
37°
e) csc x
A pendulum 20 cm long
is moved 37° from the
vertical. How much is the
lower end of the pendulum
raised by?
G
x
D
c) cot 
12.
A
C
c) tan 0 and sin 0
cos tan cot .
11.
B
satifying the inequality lies:


3
tan  −sin −cos tan 

2
2
2
, simplify the expression
sin =

E
F
13. Evaluate
tan

2
for
∈0 ,


2
if
tan =
4
3
14. Let A, B, and C be three acute angles of triangle ABC. Evaluate the following:
AB
C
a ) sin  AB−sin C
b ) cos ABcos C
c ) cos
−sin
2
2
d ) sin 2
C
AB
sin 2 

2
2
15. Complete the table below by indicating the appropriate sign of the following trigonometric ratios:

sin 
cos 
tan 
cot 
sec 
17o
csc 
+
113o
-
305o
+
-
262o
89o
168
+
o
281o
-
-
352o
16. Find the maximum and minimum values of the following trigonometric expressions:
a) A=3 cos x−1
c) C =1−sin 2 x
b) B=2−4 sin x
17. Simplify the following trigonometric expressions:
a) sin 2790 ocos 4500 ocos 7290osin 3960o
18. Evaluate
sin 2 ABcos2 B A
A=sin 2
19. Calculate
20. If
tan B=
3
5
b) sin 
, where
21 
19 
cos
sin 21 cos80 
2
2
AB=90 o

3

5
sin 2
cos2 cos2
8
8
12
12
in the isosceles triangles ABC with
21.
∣AB∣=∣AC∣
, find
cos
A
2
22.
T
r
T
A
B
B
A
C
θ r
O
O
θ
C
The ray AT is tangent to the
circle with the center O at the point T.
If AB =1, OB =4, and ∠TCO=θ,
then find tan θ = ?
In the figure above, the ray AT is tangent
to the circle centered at O with radius r at
the point A. Moreover AT ⊥ AO .
If cos θ = 3/4 , express AB in terms of r.
23. Let n∈ℤ and f : ℝ  ℝ
be a function so that
a) Draw the graph of the function f in the interval [2,3).
f  x=sin x
24. You are given the graph of
a) Use vertical stretching of the graph of
shown below, to graph
n≤xn1⇔ f  x=x−n
.
b) State whether f is periodic or not.
f  x=cos x
25. You are given the graph of
f  x=sin x ,
f  x=cos x
a) Use horizontal stretching of the graph of
x
shown below, to graph g  x=cos 
2
g  x=2sin x
x
b) Now use reflection of the graph of g  x=cos 
2
x
to graph h x=−cos 
2
c) Determine the amplitude, period, and phase shift of h x
b) Now use horizontal translation of the graph of g  x=2 sin x

h x=2 sin  x− 
4
c) Determine the amplitude, period, and phase shift of h x
to graph
f(x)=cos(x)
f( x) = sin( x)
2
2
1
1
-π/2
-5
-2π -3π/2
-π
0
π/2
π
-5
-2π -3π/2
-π
5
3π/2
2π
-π/2
0
-1
-1
-2
-2
π/2
π
5
3π/2
2π
26. Sketch the graphs of the following functions: (Use translation, reflection, compression-stretching , etc. if necessary)
Also determine the amplitude, period, and phase shift (if any) of each function.

1
y=2cos x
y=−cos x
y=2 sin x
y=sin 2 x
y=sin x
y=3sin 2 x− 
2
4
27. Find
cos x if
x∈ ,
3

2
and
1
x=arctan  
2
28. Prove the following inequalities: sin arccos x=± 1−x 2
29. Find the following functions:
sin arcsin x
cosarccos x
cosarcsin  x=± 1−x 2
tan arctan x
cot arccot x
30. Find the smallest positive value of the expressions given below if it is defined.
3 
1
a) arcsin  
2
1
b) arccos 
2
c) arcsin 
h) arcsin 1
i) arcsin   3
j) arcsin −
o) arcsin −1
p) arccos−1
2
d) arccos
3 
2
2 
k) arctan −
q) arctan 0
r) arctan 1
2
e) arcsin 
3 
3
2 
2
l) arccos0
s) arctan −1
f) arccos
2 
2
m) arcsin 0
t) arctan − 3
g) arctan   3
n) arccos1
1
u) arccos− 
2
31. Evaluate each of the given expressions by using right-angled triangle.
3
a) sin arccos 
5
4
b) tan arcsin 
5
c) sin arctan
12

5
e) cot arcsin
d) cosarctan 5
7

25
f) secarccos
15

17
32. In the figure belows, find the indicated side x (Hint: Use the Law of Sines)
A
A
45
75
x
2
x
3
30
B
C
60
B
C
33. In the triangles below, find the indicated side or angle. (Hint: Use the Law of Cosines)
A
A
30
2
4
3
B
C
x
B
θ
6
3
B
5
A
120
x
4
C
C
34. In triangle ABC, the measures a, b, and c of the sides corresponding to the angles A, B and C are given below:
 , if a 2 =b 2c 2−  3 bc
a) Find m A
c) Find m C  , if c 2=a 2b2  2 ab
35. Find the area of a triangle with sides
 , if b 2 =a 2c 2ac
b) Find m B
d) Find m C  , if c 2=a 2b 2
a=4, b=5, c=6 (Hint: Use Heron's formula)
36. In a triangle ABC, the measures a, b, and c of the sides correspond to the angles A, B and C . It is also given that
o

Find the lengths of sides b and c. (Hint: Use the Law of Tangents)
m B=105
, m C =15o , and bc= 33

2
2
37. Find A=sin a – sin b cos a – cos b , if a−b=
.
3
38. Evaluate sin ab , if sin acos b=1 and sin bcos a=  2.
39.
40.
In the figure below, ABCD is
a square with 3 DE =2 EC and
m(AEB)=θ. Evaluate tan θ.
A
D
41.
Figure below consists of
three equivalent squares.
Find tan θ if m(CAE)=θ.
A
θ
B
E
C
K
G
θ
C
12
F
θ
B
CB ⊥ BA , BC =12 as shown
in the figure.
B(0,4)
C
D
E
Find the value of
cos α if m(OAC)=α.
O
A(3,0)