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PM11
Trigonomtery Unit
Section
Topic
Assignment
SOHCAHTOA Review
Coterminal/Reference Angles
Primary Trig Ratios
Reciprocal Trig Ratios
Special Triangles & Exact Values
Solving Trig Equations
Solving Trig Equations (Part 2)
Worksheet
Worksheet
Worksheet
Worksheet
Worksheet
Worksheet
Worksheet
PM11
Lesson 1 Notes:
Review of SOH CAH TOA
Review:
opp
hyp
adj
cos A 
hyp
opp
tan A 
adj
sin A 
hypotenuse
opposite
A
adjacent
For all questions:
1.
2.
3.
Label O,A, and H on the triangle
Choose the right ratio for the information you
are given
Set up the equation and solve.
* MAKE SURE YOUR CALCULATOR IS SET TO
DEGREES *
Finding a missing side (use the sin, cos or tan button)
15
51º
x
2.4
37º
x
Finding a missing angle (use the sin-1, cos-1 or tan-1 button – “inverse sine”,
etc.)
θ
9.1
5.5
PM11
Lesson 2 Notes:
Standard, Coterminal and Reference Angles
Review:
opp
hyp
adj
cos A 
hyp
opp
tan A 
adj
sin A 
hypotenuse
opposite
adjacent
A
NEW: We can also represent angles in the 4 quadrants of the x-y coordinate
plane.
QI
Q II
terminal arm
θ
initial arm
Q IV
Q III
Standard Position:
Ex1
an angle is in standard position if…
Draw the following angles in standard position:
a) 120º
b) -130º
c) 45º
d) 400º
Coterminal Angles:
Ex2
Find 2 positive and 2 negative coterminal angles for 435º
Principal Angle:
Ex3
Find the principal angle for -520º
Reference Angle:
Ex4
Find the reference angles for:
a) 145º
b) 400º
PM11
Lesson 3 Notes:
Primary Trig Ratios
If there is a circle centred at (0,0) and if point P(x, y) is on the circle, what is the
trig ratio for that terminal arm?
I
II
P(x, y)
r
y
θ
x
y
r
x
cos  
r
y
tan  
x
2
2
x  y  r2
sin  
IV
III
When   90 , we use the reference angle of 
I
II
P(-x, y)
r
y
θ
-x
III
IV
Note: in this position all 3
trig ratios are positive
Which trig ratios are positive in each quadrant?
Ex1
Point A(-3, 2) is on the terminal arm of angle  in standard position.
Find the three primary trig ratios.
Ex2
Are the following trig ratios positive or negative?
a) sin 290º
b) sin(-35º)
c) tan 145º
d) cos (-220º)
e) tan (-100º)
f) cos 700º
PM11
Lesson 4 Notes:
1
 csc 
sin
Reciprocal Trig Ratios
1
 sec 
cos 
1
 cot 
tan
Ex1
If point B(-3, 2) is on the terminal arm of  in standard position, find
the three reciprocal trig ratios.
Ex2
If angle  is in quadrant II, and cot   
ratios.
6
, find the remaining 5 trig
5
PM11
Lesson 5 Notes:
Review:
Special Triangles & Exact Values
the 30º-60º-90º triangle has a specific ratio:
There is also a 45º-45º-90º triangle:
There are also other special angles with exact values:
Ex1
Find the exact value of the following:
a) sin 120º
b) cos 240º
d) csc 315º
e) sec(-30º)
c) tan(-135º)
Ex2
Determine the value(s) of  if 0    360 :
a) cos  1
b) sin  
3
2
c) cot    3
PM11
Lesson 6 Notes:
Review:
Solving Trig Equations
Solve: cos   
3
, 0    360
2
Now, it’s only one step away to solve this one:
(assume 0    360 unless you are told otherwise)
Ex1
2sin   3
For the next couple, you must factor:
Ex2
sin x cos x  sin x  0
Ex3
cos2 x  cos x  2  0
For this one it might be easier once you know what sec x equals, to convert
it to the primary ratio instead.
Ex4
sec 2 x 
4
3
Worksheets
PM11
Coterminal & Reference Angles
1. Find the smallest positive coterminal angle.
a. 390°
b. 420°
c. –30°
d. –405°
e. 540°
f. 830°
2. Draw each angle in standard position and find its reference angle:
a. 125°
b. 580°
c. 240°
d. 315°
e. 755°
f. –280°
PM 11
Primary Trig Ratios
Name:
The co-ordinates of a point P on the terminal arm of each  are shown. Write the
exact values of sin  , cos  , and tan  .
1.
2.
(3, 4)
(1,  3 )
Determine the exact values of sin  , cos  , and tan  if the terminal arm of angle  is in
standard position and passes through the point P.
3. P1, 6

4. P 2, 5

6. P  1,  5


7. P 3, 7

5. P 5 ,  3

8. P3,  3
State whether the value of each function is positive or negative.
9. cos 200
10. tan  225
11. sin 660
12. cos 45
13. tan 310
14. sin 120
Determine the quadrant in which  lies for each of the following:
15. tan   0, sin   0
16. cos   0, sin   0
Given that angle  is in standard position with its arm in the stated quadrant, find the
exact values of the remaining primary trigonometric ratios.
17. sin  
7
, quadrant II
25
18. tan  
3
, quadrant III
2
19. cos  
1
, quadrant IV
4
20. sin  
3
, quadrant IV
4
21. cos  
2
, quadrant II
3
PM 11
Reciprocal Trig Ratios
Name:
The co-ordinates of a point P on the terminal arm of each  are shown. Write the
exact values of sec  , csc  , and cot  .
1.
2.
(3, 4)
(1,  3 )
Determine the exact values of the six trigonometric ratios if the terminal arm of angle 
in standard position contains the given point.
3. P  7, 24

6. P 2,  5
4. P  3, 2

7. P

5. P  4, 1

8. P  1,  2 
2,  1
State whether the value of each function is positive or negative
9. cot (–260°)
10. csc 120°
11. sec(–30°)
12. csc 225°
Determine the quadrant in which  lies for the following
13. csc   0, cot   0
14. sec   0, csc  0
Given that angle  is in standard position with its terminal arm in the stated quadrant,
find the exact values of the remaining five trigonometric ratios.
3
15. sin   , quadrant II
5
3
17. cot   , quadrant III
4
2
, quadrant III
19. cot  
3
13
, quadrant IV
5
4
, quadrant IV
18. csc  
 5
16. sec  
PM 11
Special Triangles & Exact Values
Name:
Determine the exact value of the following.
1. sin 315
2. cos 270
3. tan150
4. sin 90
5. csc210
6. cot  120
7. sec120
8. sec  135
9. sin 540
10. cot 330
11. sin  150
12. csc240
Determine each value of  if 0    360 .
13. cos  
1
2
15. sin   0
17. cot  
1
3
14. tan   
1
3
16. csc  2
18. sec   
2
3
PM 11
Solving Trigonometric Equations 1
Name:
Solve for x where 0  x  360 .


4. 2 cos 2 x  cos x  0
5. sin x 2sin x  3  0

6. 2sin 2 x  3sin x  1  0
7. 2 cos 2 x  cos x  1  0
8. cos 2 x  1
3. tan x


2. tan x  3  cos x  1  0
1. 2sin x 1  0

3 tan x  1  0
PM 11
Solving Trigonometric Equations 2
Name:
Solve for  where 0    360 .
1. 1  2 cos   0
2. 2 sin  cos   2 cos   0
3. tan 2   2 tan   1
4. sin   1   sin 
5. tan 2   3  0
6. cos 2   4 cos   4  0
7. sec  csc   csc   0
8. 2 sin   1cos   1  0
9. 4 sin 2   1
10. 2 cos 2   cos   1
PM 11
Trig Review
1. Solve
ABC given: a) B  90, BC  7.2cm, and AC = 12.5 cm
b) A  34, B  90, and AB = 4.8 cm
2. Determine the principle angle ( smallest positive co-terminal angle), the reference
angle and the quadrant containing the terminal arm for:
a) 730°
b) - 235°
3. Determine the exact values of the six trigonometric ratios if the terminal arm of angle
 in standard position contains the given point:
a) P(3, -5)
b) P(  5 ,2 )
4. What quadrant has sin   0 and tan   0 ?
5. Determine cos if csc   
3
and tan   0
2
6. Given angle  in standard position with its terminal arm in quadrant lV, find the exact
3
values of the 5 remaining trig ratios is sec  
2
7. State whether the following are positive or negative
a) sin 145° b) sec 252° c) tan ( -80° ) d) csc ( -700° )
8. Determine the exact value
a) sin 45° b) sec 30° c) tan 330° d) cos 210° e) cot 480° f) csc 315°
9. Solve for x where 0  x  360
3 cot x  1
a) 2cos x 1  0
b)
c)  cos x  1 tan x 1  0
d)
e) tan 2 x  3
f) 2cos 2 x  cos x  1  0
 2cos x  1sin x 1  0
Trig Review Key:
1. a) AB = 10.2 , A  35, C  55 b) C  56 , BC = 3.24, AC = 5.79
2. a) 10,10, I b) 125,55, II
3. a) sin  
34
34
3
5
3
5
,sec  
, cot  
, csc  
, cos 
, tan  
5
3
5
3
34
34
2
 5
2
3
3
 5
b) sin   , cos  
, csc   ,sec  
, tan  
, cot  
3
3
2
2
 5
 5
4. III 5. cos  
5
3
 5
2
 5
3
3
2
, cos   , tan  
, csc  
,sec   , cot  
3
3
2
2
 5
 5
7. a) positive b) negative c) negative d) positive
3
1
2
1
1
8. a)
b)
c) 
d) 
e) 
f)  2
2
2
3
3
3
9. a) 60,300 b) 60,240
6. sin  
c)
45,180, 225
d) 90,120, 240
e) 60,120,240,300 f) 0,120, 240