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For questions 1 – 5, decide whether the equation is a trigonometric identity. Explain your reasoning.
cos2 x 1 tan2 x 1
sec x tan x 1 sin2 x sin x
csc x 2sin x 2 0
cos2 2x sin2 2x 0
sin2 csc 2 sin2 cos2
For questions 6 – 9 prove the following trigonometric identities.
sec
csc
tan
Calculation
Reason
1 sin x
1
cos x
sec x tan x
Calculation
Reason
sec x
6
3 cos x sin x
2
Calculation
Reason
sin
Prove tan
for in quadrant 1
2 1 cos
Calculation
Reason
For questions 10 – 14 solve the following equations:
sin2 x 2cos x 2
2sin 2x 2sin x 2 3 cos x 3 0
csc 4x 2 0
sin 2x sin x
Consider the equation sin2 x cos2 x
(a) Show that sin2 x cos2 x
Calculation
1 cos 4 x
8
1 cos 4 x
by filling in the table below. Start with the left hand side.
8
Reason
Calculation
Reason
(b) Find all solutions to the equation sin2 x cos2 x
2 2
.
16
Use the half-angle formulas to come up with an exact expression for each function value below. You do
not have to simplify your answers.
(a)
tan
8
(b) cos
8
tan
16
(c)
Recall that the equation for horizontal distance h in feet of a projectile with initial velocity v 0 and initial
v 2
angle is given by h 0 sin cos .
16
(d) Assume the initial velocity is 60 feet / second. What initial angle will you need to ensure that
the horizontal distance will be exactly 100 feet?
(e) Assume the initial velocity is 60 feet / second. What is the maximum horizontal distance
possible, and at what angle does this occur?
17.) Derive this identity from the sum and difference formulas for cosine:
sin a sin b
1
cos a b cos a b
2
Start with the right-hand side since it is more complex.
Answer:
Calculation
Reason
1.
Use the trigonometric subtraction formula for sine to verify this identity:
sin x cos x
2
Answer:
Calculation
Reason
2.
Find the exact value of each of the following using half-angle formulas:
a.)
b.)
c.)
d.)
Answer:
7
cos
12
17
tan
12
cot 15
sin 195
4
3. Rewrite sin x so that it involves only the first power of cosine.
Answer:
4. Prove the identity:
sin 2 x
sin x
cos 2 x
cos x
sec x
Answer:
5. Prove the identity:
cos x cos y 2 sin x sin y 2 2 2cos x y
Answer:
22.) Solve these equations graphically on the interval 0,2 . Sketch the graph and list the solutions.
(c)
sin x 1 cos x
(d) sin 2x 1 tan x
Answer:
23.) Solve these equations:
(e)
sin x sin x 1
4
4
(f)
sin x cos x
3
4
(g) tan2 x 3 tan x 2 0
Answer:

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Anonymous

Awesome! Perfect study aid.