Math 1330 Test 4 Review
ALSO do Practice Test 4
Section 5.1, 4.4, Chapters 6 & 7
ONLY THE FOLLOWING FORMULAS WILL BE PROVIDED, KNOW ALL
OTHERS.
sin (s + t) = sin s cos t + cos s sin t
cos (s + t) = cos s cos t − sin s sin t
tan s + tan t
tan (s + t) =
1 − tan s tan t
sin (s − t) = sin s cos t − cos s sin t
cos (s − t) = cos s cos t + sin s sin t
tan s − tan t
tan ( s − t) =
1 + tan s tan t
sin(2t) = 2sin t cos t
cos(2t) = cos 2 t − sin 2 t
sin
1 − cos s
s
=±
2
2
cos
1 + cos s
s
=±
2
2
tan
sin s
s
=
2 1 + cos s
1. Which of the following is equivalent to:
a. cos(α + 2π ) sec(−α ) − sin(−α ) csc (α − 4π )
A. 1
B. 0
Math 1330 Test 4 Review
C. -1
D. -2
E. 2
1
b.
sec α
1 + sin α
−
1 − sin α
cos 3 α
A. 2 sec 2 α B. 0 C. sec 2 α
1 − cot 2 α
c.
+ 2 cos 2 α
1 + cot 2 α
A. 1
B. 0
Math 1330 Test 4 Review
D. 2 csc 2 α
C. − 1 + 2 cos 2 α
E. 2 cos 2 α
D. 2
E. 2 sin 2 α
2
d.
1 + sec α
1 − sec α
cos α + 1
C. tan α
D. sec α
E. cos α
cos α − 1
2. Given sin (x) = 5⁄7 with 0o < x < 90o and sin (y) = -3⁄4 with 180o < y < 270o.
Find cos( x − y ) .
Recall: cos( x − y ) = cos x cos y + sin x sin y
A. sin α
B.
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3. Given sec (x) = -5 with 90o < x < 180o. Find sin (2x).
Recall: sin(2 x) = 2sin x cos x
1
1 
with 90o < x < 180o. Find cos x  .
8
2 
1 + cos x
 x
Recall: cos   = ±
and 90o < x < 180o.
2
2
 
4. Given cos x = −
Math 1330 Test 4 Review
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 9π 
5. Use the half-angle formula for sine to find the exact value of sin 
.
 8 
1 − cos θ
θ 
Recall: sin   = ±
2
2
9π
 9π 
a. Which quadrant does
live in?
b. Is sin 
 positive or
8
 8 
negative?
θ 
 9π 
c. Rewrite sin 
 so that it’s in the form sin   and then calculate.
 8 
2
What does θ equal?
6. Find sin(105o ) by using the sum formula for sine.
Recall: sin( x + y ) = sin x cos y + cos x sin y
Math 1330 Test 4 Review
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7. Solve 3sin 2 x + 2sin x − 5 = 0 in the interval [ 0 , 2π).
8. Determine all solutions to cos(2 x) =
1
on the interval [ 0 , 2π).
2
k=
k=
k=
k=
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9. How many solutions does each of the following equations contain…
1
2
a. ( cos x ) = in [0, 2 π )?
4
b. sin(2 x) = −9 cos x in [- π , π ]?
Hint: Use sin(2 x) = 2sin x cos x
c. cos x sin x + 3cos x + sin x + 3 = 0 in [0, 2 π )?
Hint: Factor by grouping.
Math 1330 Test 4 Review
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10. Find all solutions for each of the following equations in the indicated interval.
 π 5π 
a. sec 2 x = 1 in  − , 
 2 2 
π

 1 4
b. sin  π x +  = 1 in  − , 
5

 5 5
Math 1330 Test 4 Review
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6
1
in and sin x = . Find the
4
5
area of ∆ ABC. Hint: The double angle formula for sine will be useful.
sin ( 2 x ) = 2sin x cos x
11. In ∆ ABC, the measure of ∠A = (2 x) o , c = 5 in, b =
12. To measure the height of a cloud, you place a bright searchlight directly below the
cloud and shine the beam straight up. From a point 100 feet away from the searchlight,
you measure the angle of elevation of the cloud to be 83°. How high is the cloud?
13. An isosceles triangle has sides 3 cm, 3 cm and 1 cm. Give the tangent of a base
angle of this triangle.
14. Find the area of triangle XYZ if ∠ Y = 30° , z = 10 mm and x = 6 mm.
1
Recall: A = ab sin θ
2
15. Find the area of a regular quadrilateral inscribed in a circle of radius 2cm.
Math 1330 Test 4 Review
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16. In triangle POM angle P = 30 o , angle O = 45 o , and MO = 16 cm. Find PM.
5
3 m. The measure of angle A is
3
30 o . How many choices are there for the measure of angle C?
17. Given triangle ABC with AB = 5 m and BC =
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18. Given ∆ PEZ, p = 6 cm, e = 13 cm, and z = 11 cm. Find cos(Z).
19. Two cyclists leave the corner of State Street and Main Street simultaneously. State
Street and Main Street are not at right angles; the cyclists' paths have an angle of 135o
between them. How far apart are the cyclists after they each travel 3 miles?
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20. An isosceles triangle has an apex angle measuring 150 o . The two equal sides have
length Q inches. What is the length of the base of the triangle?
Math 1330 Test 4 Review
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