-1-#
Write each positive angle in degrees for each terminal side between 0° and 360°
NAME ____________________________________
Standard Position
Initial Side
Terminal Side
Coterminal Angles
Positive Angles Negative Angles
-2-#
Draw each angle in standard form.
Determine the quadrant in which the terminal side lies.
190° 1° 300°
405° -90° 135°
-330° 179° -60°
-3-#
Write each positive angle in RADIANS for each terminal side between 0 and 2π
MEASURING ANGLES IN RADIANS
When a measurement of
an angle is given with no
units, the angle is
measured in radians
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Draw an angle
measuring 6.28 radians
Draw an angle
measure of 5
-4-#
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Put all the information together on one circle
Label each terminal side in degrees.
Label each terminal side in radians.
-5-#
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-6-#
Trigonometric Functions in Right Triangles
Special Right Triangles
Complete the following using the triangles above:
sin
cos
tan
θ
θ
θ
=
=
=
csc
sec
cot
θ
θ
θ
=
=
=
sin 45
cos 45
tan 45
csc 45
sec 45
cot 45
°=
°=
°=
°=
°=
°=
sin30
cos30
tan 30
csc30
sec3 0
cot30
°=
°=
°=
°=
°=
°=
sin 60
cos 60
tan 60
csc60
sec 60
cot 60
°=
°=
°=
°=
°=
°=
-7-#
Write the angle measurements in RADIANS. Then fill in the lengths of the sides.
Complete the following using the triangles above:
Find the value of ANY trig ratio by drawing a triangle with the hypotenuse as the terminal
side of the given angle.
sin
4
cos
4
tan
4
csc
4
sec
4
cot
4
π
π
π
π
π
π
=
=
=
=
=
=
sin
3
cos
3
tan
3
csc
3
sec
3
cot
3
π
π
π
π
π
π
=
=
=
=
=
=
sin
6
cos
6
tan
6
csc
6
sec
6
cot
6
π
π
π
π
π
π
=
=
=
=
=
=
-8-#
Find the exact value of each trigonometric ratio. Draw a triangle to help you determine
the ratio. Leave answers in simplified radical form.
1.
cos
4
π
=
2.
cos
3
π
=
3.
sec
3
π
=
4.
sin
6
=
π
5.
tan
4
π
=
6.
cos
6
π
=
7.
sec
6
=
π
8.
tan
6
=
π
9.
cos
6
=
π
10.
csc
4
=
π
11.
tan
3
π
=
12.
sec
6
=
π
13.
cot
6
=
π
14.
sin
3
=
π
15.
cot
4
=
π
16.
csc
3
=
π
17.
tan
6
π
=
18.
sec
6
π
=
19.
cos
6
=
π
20.
tan
3
=
π
21.
cos
4
=
π
22.
csc
6
=
π
23.
sec
4
=
π
24.
cot
3
π
=
25.
sin
4
π
=
26.
sec
3
=
π
27.
sin
3
π
=
28.
tan
3
π
=
-9-#
Reference Angles –
Find
5
cos
4
π
Find
5
csc
6
π
−
Find
7
tan
4
π
Find
13
cot
6
π
Find
3
sin
4
π
Find
sec
3
π
−
Find
2
tan
3
π
1. Draw the angle in standard position.
2. Create a triangle - draw a vertical line from
the terminal side (hypotenuse) to the x-axis
3. Find the acute angle formed - it may be a part
of the given angle or it may be outside of the
given angle. This is called the reference angle
4. Write in the lengths of the sides based on the
reference angle formed (use special right
triangles)
Hypotenuse - ALWAYS positive
x or y side - may be positive or negative
5. Find the ratio - simplify if necessary.
-10-#
Find the exact value of each trigonometric ratio. Draw a sketch of your angle, complete
the reference triangle in the correct position, and label all sides of the triangle.
1. sin
6
5
π
= 2. csc
5
3
π
=
3. cos
7
6
π
=
4. tan
5
4
π
=
5. cot
3
2
π
= 6. cos
6
11
π
=
7. sec
3
5
π
= 8. sin
4
3
π
=
9. tan
5
6
π
= 10. cos
2
3
π
=
11. sin
3
π
= 12. cos
7
4
π
=
13. csc
3
4
π
= 14. sec
3
π
=
-11-#
Which trig ratios are positive in the first quadrant?
Which trig ratios are positive in the second quadrant?
Which trig ratios are positive in the third quadrant?
Which trig ratios are positive in the fourth quadrant?
Quadrantal Angles
-12-#
Quadrantal Angles - Find the exact value of each trigonometric ratio. Draw the angle and mark
the point on the terminal side.
1. sin
2
π
= 2. cos
3
2
π
=
3. tan
π
=
4. sec
2
π
=
5. csc
3
2
π
=
6. cot
π
2
=
7. tan
2
π
=
8. sin
π
=
9. cot 0 = 10. csc
2
π
=
11. cos
π
= 12.
sec
3
2
π
=
13. cos
2
π
= 14. sin
2
3
π
=
-13-#
Review Reference Angles
#
1.
cos
4
π
=
2.
cos
3
π
=
3.
sec
3
π
=
4.
tan
4
π
=
5.
5
sin
3
π
=
6.
sin
2
π
=
7.
cos
6
π
=
8.
5
sec
6
π
=
9.
sin
6
π
=
10.
cos
π
=
11.
7
cos
6
π
=
12.
7
csc
4
π
=
13.
tan
3
π
=
14.
7
sec
6
π
=
15.
7
sin
4
π
=
16.
7
tan
6
π
=
17.
2
cos
3
π
=
18.
3
sec
2
π
−
=
19.
5
csc
3
π
=
20.
3
cot
4
π
=
21.
tan
6
π
=
22.
sec
6
π
=
23. sin
3
2
π
=
24.
2
tan
3
π
=
25.
5
cos
6
π
=
26.
5
cos
4
π
=
27.
tan
π
=
28.
sec
2
π
−
=
29.
cot
3
π
=
30.
11
csc
6
π
=
31.
sin
4
π
=
32.
3
cot
2
π
−
=
33.
sin
3
π
=
34.
sin 2
π
=
35.
tan
3
π
=
-14-#
Find the exact value of the following trigonometric ratios.
-15-#
-16-#
Given a point or a ratio, find all trig values!
-17-#
-18-#
Quadrant Restrictions
-19-#
For the following problems, determine in which quadrant the angle must lie. Then find the
other trigonometric ratio.
1. Given: sin
θ
=
4
1
−
and cos
θ
> 0. Find tan
θ
.
2. Given: tan
θ
=
4
3
and sin
θ
< 0. Find cos
θ
.
3. Given: cos
θ
=
4
3
and sin
θ
> 0. Find csc
θ
.
4. Given: tan
θ
= -
12
5
and sin
θ
> 0. Find cos
θ
.
5. Given: sec
θ
= -3 and tan
θ
> 0. Find cot
θ
.
6. Given: csc
θ
=
2
and tan
θ
< 0. Find cos
θ
.
-20-#
For the following problems, a point on the terminal side of angle
θ
is given. Find the exact value of
the other trig ratio.
7. (-4, 3); cos
θ
8. (-2, -5); tan
θ
9.
( )
2,2 −
; sin
θ
Find the exact value of each trigonometric function or find
θ
in the interval
πθ
20 ≤≤
.
10.
7
sin
4
π
=
12.
3
cos
2
θ
= −
13.
5
cot
3
π
=
14.
csc2
π
=
15.
tan
θ
=
undefined 16.
7
cos
6
π
=
17.
4
sin
3
π
=
18.
3
sec
4
π
=
19.
tan 1
θ
=±
20.
11
sin
6
π
=
21.
1
sin
2
θ
=
22.
1
cos
2
θ
= −
-21-#
