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Review Problems Quarterly #3 Exam
Pre-Calculus
2017
1.
Solve for all possible values of x within the domain of
. Do not use your graphing calculator.
a)
b)
c)
d)
e)
f)
2.
A ladder 3.0 m long is placed against a building at an angle of 60º with the ground. How high up the
wall will the ladder reach?
3.
To find the height of a pole, a surveyor moves 100 feet from the base of the pole and then, from an eyelevel height of 6.5 feet, measures the angle of elevation to the top of the pole to be 40º. Find the height
of the pole to the nearest foot.
4.
5.
The cable supporting a ski lift rises 3 feet for every 8 feet of horizontal length. The top of the cable is
fastened 675 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of
Evaluate each:
a.)
b.)
c.)
d.) arccos(-.7398)
e.) arctan(1.5)
f.) arcsin(2.3)
g.)
h.)
.
6.
Find the amplitude, period, phase shift, and vertical shift of each function.
a.)
7.
b.)
c.)
Amp = _________
Amp = _________
Amp = _________
Period = ________
Period = ________
Period = ________
VS = __________
VS = __________
VS = __________
PS = __________
PS = __________
PS = __________
A windmill has a radius of 27 ft and its center is 52 ft above the ground. The wheel rotates at a constant
angular velocity of
radians per minute. The height of a point on the windmill’s blade as a function
of time is given by the equation:
, where h is the height in feet and t is the time
in minutes. Find the approximate value of h(t) when t = 7 minutes.
Chapter 5: Analytic Trigonometry
Fundamental Identities
Pythagorean Identities
2
2
sin x + cos x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Reciprocal Identities
1
csc x =
sin x
1
sec x =
cos x
1
cot x =
tan x
Quotient Identities
sin x
cos x
tan x =
cot x =
cos x
sin x
Identities for Negatives
sin (-x) = - sin x
cos (-x) = cos x
tan (-x) = - tan x
8. Simplify:
a.) sec x • cos x
sin 2 x
+ cos x
b.) cos x
c.) cos u + sin u • tan u
d.)
sin t
cos2 t −1
e.)
csc x
cot x + tan x
f.)
sin x
1 + cos x
−
1 − cos x
sin x
Verifying Trigonometric Identities
Hints:
1.) Work on one side of the equation only. Choose the more complicated side.
2.) If you have the sum or difference of fractions, try finding a common denominator.
3.) Try algebraically manipulating one side (i.e. factor, separate fractions, multiply, etc.).
Verify each identity.
9. (sec x)(sin x + cos x) = tan x + 1
1− (cos x − sin x)2
= 2sin x
cos x
10.
cos A − sin A
= csc A − sec A
11. sin A cos A
3cos 2 A + 5sin A − 5 3sin A − 2
=
cos 2 A
1+ sin A
12.
Sum, Difference, and Cofunction Identities
sin (x + y) = sin x cos y + cos x sin y
sin(x – y) = sin x cos y – cos x sin y
cos (x + y) = cos x cos y – sin x sin y
cos(x – y) = cos x cos y + sin x sin y
tan x + tan y
tan (x + y) = 1− tan x tan y
tan x − tan y
tan (x – y) = 1+ tan x tan y
sin(90° – x) = cos x
cos (90° – x) = sin x
tan (90° – x) = cot x
cot (90° – x) = tan x
sec (90° – x) = csc x
csc (90° – x) = sec x
Evaluate exactly using Sum and Difference Identities.
13. sin 75°
14. cos74° cos44° + sin74° sin 44°
15. Evaluate cos 105°
π
5
16.
π
1 + tan π tan
5
Verify:
17.
cot x − tan y =
cos(x + y)
sin x cos y
tan π − tan
18. tan A + tan B = sin(A + B)
tan A − tan B
sin(A − B)
Double-Angle Identities
sin 2x = 2 sin x cos x
2
2
cos 2x = cos x – sin x
= 1 – 2sin2 x
= 2 cos2 x – 1
tan 2x =
2 tan x
1− tan 2 x
Verify:
2 tan x
19.
= sec 2 x
sin 2x
20. (sin x – cos x)2 = 1 – sin 2x
cos 2t
1+ tant
=
21. 1− sin 2t 1− tant
Trigonometric Equations
Steps to Solve Trigonometric Equations:
1.) If only one trig function exists, isolate it and solve for the variable.
2.) If more than one trig function exists, try to factor out a common factor or substitute with identities.
3.) If two or more different trig. functions exist, try to use identities to rewrite in terms of one trigonometric
function.
4.) If variables have different coefficients, try rewriting as the same variable (i.e. If sin 2x = sin x, rewrite
sin 2x as 2sin x • cos x)
5.) Solve by factoring or by quadratic formula.
Find exact solutions over the indicated interval.
22.
cos x =
− 3
2
0 ≤ x ≤ 360!
24. 2sin 2x = 3
1
cos 2 x = sin 2x
2
26.
Estimate the solutions.
28. tan x = -0.84
0<x<p
0 ≤ x ≤ 360!
−π
π
≤x≤
2
2
23. sin x =
1
2
all reals
25. 2 cos2 x + cos x = 1
27. 6sin2 x + 5 sin x = 6
0 < x < 2p
0° < x < 90°