Download Final Exam Name. . MATH 150 SS Number. . Trigonometry Rec

Final Exam
MATH 150
Trigonometry
Name.
SS Number.
Rec. Instructor.
.Hour.
.
.
.
16 May 2003
General Instructions: Show all work. If you must continue your work outside the space
provided, please give instructions in the space provided as to where the continuation is found.
You may use calculators and a 3-by-5 card with notes on one side only. Put your name,
instructor’s name and recitation hour on the back of the note card.
Question Point Value Points Earned
1.
12
2.
12
3.
12
4.
12
5.
12
6.
12
7.
12
8.
12
9.
12
10.
12
11.
12
12.
12
13.
12
14.
12
15.
20
16.
12
1
1. Suppose sin t = − 35 and cot t > 0. Find the values of all six trigonometric functions of
the angle t.
2
2. Find the exact values of all six trigonometric functions of θ if θ is the (measure of) an
angle in standard position with the point P = (7, −9) on its terminal side.
3
3. Verify the trigonometric identity
cos4 θ − sin4 θ = cos 2θ
4
4. Find all solutions to the trigonometric equation
sin2 x − sin x = 1
5
5. Suppose α is a second quadrant angle with sin α =
5
with cos β = 13
. Find the following
a) The quadrant containing α + β
b) An exact value for sin(α + β)
c) An exact value for cos(α + β)
6
4
5
and β is a fourth quadrant angle
6. Find exact values for sin( 2θ ), cos( 2θ ), and tan( 2θ ) if sin θ = − 12
and 180◦ < θ < 270◦ .
13
7
7. Use a sum-to-product formula to solve
sin 3x = sin 5x
give all solutions in the interval [0, 2π). Approximate all solutions in radians correct to three
places after the decimal.
8
8. Find exact values for the following:
a) sin( 21 arccos 12 )
9
b) tan(arcsin − 41
)
9
9. Solve the (standardly labeled) triangle 4ABC with a = 4, b = 9, c = 7. (Give angles in
degrees correct to the nearest .01◦ .)
10
10. Solve the (standardly labeled) triangle 4ABC with α = 15◦ , β = 50◦ , a = 8. (Give
lenghts correct to the nearest .1 and angles in degrees correct to the nearest .1◦ .)
11
11.
a) Find the trigonometric form of the complex number z = −2 + 2i.
b) Use DeMoivre’s theorem to find z 4 . Give your answer in the form a + bi.
c) Use DeMoivre’s theorem to find all cube roots of z. Leave your answers in trigonometric
form.
12
12. Find the equation of the parabola with focus (2, −8) and directrix y = 10
13
13.
a) Find the vertices, center, and foci of the ellipse whose equation is
(x − 2)2 (y + 3)2
+
=1
25
169
b) Sketch the ellipse of part a), showing the foci.
14
14.
a) Find the vertices, center and foci of the hyperbola whose equation is
x2 y 2
−
=1
9
16
b) Sketch the hyperbola of part a), showing the foci and asymptotes.
15
15. Each of the following equations is the equation of a conic section, those involving
x and y are given in rectangular coordinates, those involving r and θ are given in polar
coordinates. Classify each as an ellipse, parabola, or hyperbola. Show enough work to
justify your classification.
a)
3
r=
2 + 2 sin θ
b)
x2 − y 2 − 2x − 16y + 16 = 0
c)
r=
3
2 − 3 cos θ
d)
y = 6x2 − 24x + 20
e)
r(3 + sin θ) = 15
16
16.
a) Find an equation in general form for the equation of the line through the point (3,-1)
and parallel to the line x = 5.
b) Find an equation in general form for the equation of the line through the point (3, -1)
and perpendicular to the line 2x − 6y = 5
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