Download Advanced Trigonometry Problems

Advanced Trigonometry Problems
1.
Right triangle ABC has angle measure , , and  degrees and side lengths a, b,
and c inches, as illustrated below. Which of the following is true about the value of
the product tan  tan  ?
F. The value is 1.
G. The value is
b2
c2
.
b2c2
H. The value is ca4 .
J. The value is undefined.
K. The value cannot be determined from the given information.
2.
If the sine of an angle is 0.8 and its cosine is –0.6, what is its cotangent?
F. - 53
G. - 43
H. - 34
1
J.
5
K. 54
3.
Which of the following is a consequence of the identity
sin 2x = 2(sin x)(cos x)
F.
G.
H.
J.
K.
sin y = (sin 12 y)(cos 12 y)
sin 3y = 3(sin 2y)(cos y)
sin 4y = 2(sin 2y)(cos 2y)
sin 4y = 4(sin 2y)(cos 2y)
sin 8y = 2(sin 4y)(cos 2y)
4.
If cos x =
1
2
and  < x < 2, what is the value of sin x ?
F. - 23
G. - 12
H. 32
3
J.
4
K.
5.
3
2
If tanA = q, which of the following expressions must also equal q ?
F. cot A
G. sec A – 1
H. sin A + cos A
sin A
J. cos
A
A
K. cos
sin A
6.
Compared to the graph of cos , the graph of 4 cos 2 has:
F. 8 times the amplitude and the same period.
G. 4 times the amplitude and half the period.
H. 4 times the amplitude and twice the period.
J. 14 the amplitude and half the period.
K.
7.
1
4
the amplitude and twice the period.
For all values of x where sin x, cos x and tan x are all defined,
(sin x)(cos x)(tan x) = ?
F. sin2x
G. cos2x
H. tan2x
J.
1
K. –1
8. Where is the terminal side of an angle in standard position whose measure is - 116 
radians? (Note: an angle in standard position has its vertex at the origin and initial
side extending along the positive x-axis. Positive angles are measured in a counterclockwise direction from the initial side to the terminal side. Shown below is an
example of an angle measuring + 34  radians, in standard position, where the terminal
side is in quadrant II.)
A.
B.
C.
D.
E.
Along the negative y-axis
In quadrant I
In quadrant II
In quadrant III
In quadrant IV
9. Compared to the graph of y = sin , the graph of y = 2 sin  has:
F.
G.
H.
J.
K.
Twice the period and the same amplitude.
Half the period and the same amplitude.
Twice the amplitude and half the period.
Half the amplitude and the same period.
Twice the amplitude and the same period.
10. Two lookout towers are located about 7 miles apart, at the same elevation. A fire is
sighted at angles of 37 and 42 from the line of sight between the towers, as
indicated in the diagram below. Which of the following expressions, if any, gives the
approximate distance, in miles, between the fire and tower A ?
(Note: The law of sines states that the ratio between the length of the side opposite an
angle and the sine of that angle is the same for all interior angles in the same triangle)
F.
G.
H.
J.
K.
The distance cannot be approximated without more information.
37 2  42 2
7 cos 42
cos101
7 tan 42
7 sin 42
sin101
11. A surveyor took and recorded the measurements shown in the figure below. If the
surveyor wants to use these 3 measurements to calculate the length of the pond,
which of the following would be the most directly applicable?
F.
G.
H.
J.
K.
The Pythagorean theorem
A formula for the area of a triangle
The ratios for the side lengths of 30-60-90 triangles
The ratios for the side lengths of 45-45-90 triangles
The law of cosines: For any ABC, where a is the length of the side opposite
A, b is the length of the side opposite B, and c is the length of the side
opposite C, a2 = b2 + c2 – cos(A)
12. In ABC, shown below, the measure of B is 41, the measure of C is 34, and
AB is 25 units long. Which of the following is an expression for the length, in units,
of BC ?
(Note: The law of sines states that, for any triangle, the ratios of the sines of the
interior angles to the lengths of the sides opposite those angles are equal.)
A.
B.
C.
D.
E.
25 sin105
sin 41
25 sin105
sin 34
25 sin 75
sin 41
25 sin 41
sin105
25 sin 34
sin 75