Download PRECAL. TEST 2 REVIEW

PRECAL. TEST 2 REVIEW
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
1) The equation θ2 = tan-1 (ωC/G) gives the phase angle of impedance in the parallel portion of a distributed
constant circuit. Find θ2 if ω = 150 radians per second, C = 0.08 μF per kilometer, and
G = 1.85 μsiemens per kilometer.
Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
2) sin x - sin x cosx = sin3 x
3) cos (x + π) = cos x
Complete the identity.
4)
(sin x + cos x)2
=?
1 + 2 sin x cos x
Verify the identity.
5) cot θ ∙ sec θ = csc θ
6) csc2 u - cos u sec u= cot2 u
7) (1 + tan2 u)(1 - sin2 u) = 1
8) cot 2 x + csc 2 x = 2 csc 2 x - 1
Find the exact value of the expression.
9) cos (165°) cos (45°) + sin (165°) sin (45°)
Use the given information to find the exact value of the expression.
4
2
10) sin α = , α lies in quadrant II, and cos β = , β lies in quadrant I
5
5
Find cos (α - β).
Verify the identity.
cos(α + β)
11)
= cot β - tan α
cos α sin β
Use the given information to find the exact value of the expression.
7
2
12) sin α =
, α lies in quadrant II, and cos β = , β lies in quadrant I
25
5
Verify the identity.
π
tan x - 1
13) tan x =
4
1 + tan x
1
Find cos (α - β).
14) cos 4θ = 2 cos2 (2θ) - 1
Solve the problem.
15) If a projectile is fired at an angle θ and initial velocity v, then the horizontal distance traveled by the projectile is
1 2
given by D =
v sin θ cos θ. Express D as a function of 2θ.
16
Use a half-angle formula to find the exact value of the expression.
3π
16) sin
8
Use the given information to find the exact value of the trigonometric function.
6
θ
17) csc θ = - , tan θ > 0
Find cos .
5
2
Express the product as a sum or difference.
11x
x
18) cos
cos
2
2
Express the sum or difference as a product.
19) sin 75∘+ sin 15∘
Verify the identity.
sin α - sin β
α-β
α+β
20)
cot
= tan
sin α + sin β
2
2
Solve the equation on the interval [0, 2π).
3
21) cos 2x =
2
22) sin2 x - cos2 x = 0
23) tan x + sec x = 1
Solve the equation on the interval [0, 2π).
24) -tan2 x sin x = -tan2 x
Find all solutions of the equation.
25) 2 cos x - 1 = 0
Solve the equation on the interval [0, 2π).
26) sin 2x + sin x = 0
Rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than
1.
27) 2 sin2 x cos2 x
2
Find the exact value under the given conditions.
4 π
21
3π
28) cos α = - ,
, π<β<
< α < π; sin β = 5 2
5
2
3
Find tan (α + β).
Answer Key
Testname: TST2412_REVIEW2
1) 80.2°
π
2)
4
20)
3) 0
4) 1
cos
cos θ
1
1
5) cot θ ∙ sec θ =
∙
=
= csc θ
sin θ cos θ sin θ
6) csc2 u - cos u sec u = csc2 u - cos u ∙
7) (1 + tan2 u)(1 - sin2 u) = sec2 u ∙ cos2 u =
α+β
2
α+β
sin
2
1
= csc2 u cos u
1 = cot2 u
sin α - sin β
=
sin α + sin β
1
∙ cos2 u
cos2 u
=1
π 3π 5π 7π
,
,
,
4 4
4
4
π
5π
+ 2nπ or x =
+ 2nπ
3
3
26) 0,
11)
cos(α + β)
cos α cos β - sin α sin β cos α cos β
=
=
cos α sin β
cos α sin β
cos α sin β
27)
sin α sin β cos β
sin α
=
= cot β - tan α
cos α sin β
sin β
cos α
1 1
- cos 4x
4 4
28)
-6 + 4 21
8 + 3 21
π
tan x - tan π/4
tan x - 1
.
=
=
4
1 + (tan x)(tan π/4) 1 + tan x
14) cos 4θ = cos[2(2θ)] = 2 cos2 (2θ) - 1
1 2
15) D =
v sin 2θ
32
16)
1
2
19)
2
18 - 3 11
6
17) 18)
2+
1
(cos 5x + cos 6x)
2
6
2
4
α-β
α+β
cot
2
2
22)
-6 + 4 21
25
13) tan x -
α+β
α-β
cos
2
2
π 11π 13π 23π
,
,
,
12 12 12 12
25) x =
-48 + 7 21
125
2 sin
= tan
10)
12)
α-β
α+β
cos
2
2
21)
23) 0
24) 0, π
8) cot 2 x + csc 2 x = csc 2 x - 1 + csc 2 x = 2 csc 2 x - 1.
1
9) 2
2 sin
2π
4π
, π,
3
3
sin
α-β
2
cos
α-β
2
=
∙