Trigonometry Review
Trigonometry Review
1. Sketch [pic] of:
a) y = 2 sin3[pic]-1 b) y = -4 cos[pic]
c) y = [pic]sin2[pic] d) y = -3 cos[pic]-2
2. Sketch f([pic]) = 2 sin [pic]on the interval [pic].
a) Find the average rate of change (to 5 decimals) of the function from [pic]. What does this
tell us about the graph?
b) Find the instantaneous rate of change (to 5 decimals) of the function at [pic] What does this tell us about the graph?
3. State two co-terminal angles for each of the following. Your answers must include [pic]
a) [pic] b) [pic] c) [pic] d) [pic]
4. Find each function value.
a) csc[pic] if sin[pic] b) cot [pic] if [pic]
5. Find each of the following. Use exact values only.
a) sin[pic] b) sec [pic] c) cot [pic] d) cos [pic]
6. Find all values for [pic].
a) cos [pic] b) sec[pic] c) cot [pic] d) tan[pic]
7. Solve for[pic].
a) cos[pic] b) sin[pic] c) tan [pic]
d) cos [pic] e) sin[pic] f) cos [pic]
8. Solve for[pic].
a) cos2[pic] b) 2sin2x + sin x – 1 = 0 c) 10cos2(2x) + 7cos(2x) = 6
d) 4cos2(2x) – 1 = 0 e) 3tan2x = 1 f) 2tan2x + tan x – 3 = 0
g) 6cos2x – sin x – 4 = 0 h) 2cos x = 1 – sin2 x i) 2sin2 x = -cos x + 1
9. a) Convert 330( to radians b) Convert [pic]radians to degrees.
10.Use compound angles to determine the exact value for
a) sin[pic] b) cos [pic]
11. Write as a single trig function.
a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic][pic]
12. Evaluate the following.
a) sin 45º cos15º - cos 45º sin 15º b) [pic]
c) cos215º + sin215º d) sin [pic]
e) cos215º - sin2 15º f) (cos 15º + sin 15º)2
g) sin[pic]
13. Simplify.
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
14. Develop a formula for [pic]in terms of [pic]
15. If [pic] and [pic], where [pic] and [pic], find the following.
a) sin(x – y) b) cos(x+y) c) sec(x – y)
16. If [pic] for [pic] find the following.
a) cos 2x b) csc 2x c) sin 4x
17. Prove the following identities.
a) [pic] b) (sin x – cos x)2 = 1 – 2sin x cos x
c) cos3x + cos x sin2x =[pic] d) (1 + sec x)(1 – sec x) = -sin2x sec2x
e) [pic] f) tan x +1 = [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
k) [pic]
18. Sketch y = cot x for [pic].
19. a) State the equations of the vertical asymptotes for
i) y = sec x for [pic]
ii) y = tan x for [pic]
b) State the local minimums for y = csc x for [pic].
c) State the period of y = tan 2x.
Answers
1. (a) (b)
(note: vertical scale is 1, horizontal scale is [pic] ) (note: vertical scale is 1, horizontal scale is [pic])
c) (d)
(note: vertical scale is 1, horizontal scale is [pic]) (note: vertical scale is 1, horizontal scale is [pic])
2.
(note: vertical scale is 1, horizontal scale is [pic])
a) ARC = -0.95493 –This tells us the slope of the secant between the points [pic].
b) IRC = -1.73726 – This tells us the slope of the tangent at [pic].
3. a) [pic] b) [pic] c) [pic] d) [pic]
4. a) [pic] b) [pic]
5. a) [pic] b) -2 c) [pic] d) [pic]
6. a) [pic] b) [pic] c) [pic] d) [pic]
7. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] ,[pic] f) [pic]
8. a) [pic] b) [pic] c) [pic] d) [pic]
e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]
9. a) [pic]radians b) 32.7(
10. a) [pic] b) [pic]
11. a) sin (A – B) b) cos(3A) c) [pic] d) cos(6M) e) 5 sin(2x)
f) -2sinx
12. a) [pic] b) [pic] c) 1 d) -[pic] e) [pic] f) [pic] g) [pic]
13. a) 2 b) sin4x c) [pic] d) 2sina e) 1 f) 1
14. [pic] 15. (a) [pic] (b)[pic] (c) [pic]
16. (a) [pic] (b) [pic] (c) [pic]
18.
-----------------------
19. (a) (i) x = (2n+1)[pic] (ii) x = (2n+1)[pic] (b) local minimums at (x,1) where [pic] (c) Period is [pic]
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