Mock Final
Math 1060-002
ID: U
Name:
Question:
1
2
3
4
5
6
7
8
9
10
Total
Points:
10
10
10
10
10
10
10
10
10
10
100
Score:
• Useful formulas:
• cos
• sin(u + v) = sin u cos v + cos u sin v
• tan u2 =
• sin(u − v) = sin u cos v − cos u sin v
• cos(u + v) = cos u cos v − sin u sin v
• cos(u − v) = cos u cos v + sin u sin v
• tan(u + v) =
tan u+tan v
1−tan u tan v
• tan(u − v) =
tan u−tan v
1+tan u tan v
• sin 2u = 2 sin u cos u
• cos 2u = cos2 u − sin2 u = 2 cos2 u − 1 =
1 − 2 sin2 u
• tan 2u =
2 tan u
1−tan2 u
u
2
=±
q
1+cos u
2
1−cos u
sin u
=
• sin u sin v = 12 [cos(u − v) − cos(u + v)]
• cos u cos v = 21 [cos(u − v) + cos(u + v)]
• sin u cos v = 12 [sin(u + v) + sin(u − v)]
• cos u sin v = 21 [sin(u + v) − sin(u − v)]
cos u−v
• sin u + sin v = 2 sin u+v
2
2
• sin u − sin v = 2 cos u+v
sin u−v
2
2
u−v
cos
• cos u + cos v = 2 cos u+v
2
2
u−v
• cos u − cos v = −2 sin u+v
sin
2
2
• sin2 u =
1−cos 2u
2
• cos A =
b2 +c2 −a2
2bc
• cos2 u =
1+cos 2u
2
• cos B =
c2 +a2 −b2
2ac
• tan2 u =
1−cos 2u
1+cos 2u
• cos C =
a2 +b2 −c2
2ab
• sin u2 = ±
q
1−cos u
2
sin u
1+cos u
• proj v u =
u·v
kvk2
v
Page 2
1. (a) 5 points Find the radian measure of the central of a circle of radius 80 cm that
intercepts an arc of length 160 cm.
(b) A circular power saw of radius 10 centimeters rotates at 5000 revolutions per minute.
i. 3 points Find the angular speed of the saw blade in radians per minute.
ii. 2 points Find the linear speed of the saw blade in meters per minute.
[ 1 meter = 100 centimeters ]
Page 3
2. Sketch the graph of the following functions.
(a) 5 points y = sin x − π4
(b) 5 points y = tan x2
3. Evaluate
(a) 5 points tan arcsin − 53
Page 4
(b) 5 points If tan θ = − 43 , find the value of the rest of the trigonometric functions.
4. 10 points A cellular telephone tower that is 150 meters tall is placed on the top of a
mountain that is 1200 meters above the sea level. What
√ is the angle of depression from
the top of the tower to a cell phone user who is 250 3 horizontal meters away and 600
meters above the sea level?
Page 5
5. Verify the following identities.
(a) 5 points
cos t+cos 3t
sin 3t−sin t
(b) 5 points sec
u
2
= cot t
=±
q
2 tan u
tan u+sin u
Page 6
6. Solve the following trigonometric equations. Write the general solutions for each of the
problem.
(a) 5 points tan 2x − 2 cos x = 0
(b) 5 points sec2 x + tan x − 3 = 0
Page 7
7. 10 points Given tan u = − 43 ,
3π
2
< u < 2π, find the exact value of sin u2 , cos u2 , and tan u2 .
8. (a) 5 points Solve the triangle(s) which has a = 1, b =
exist, find both.
Page 8
√
3, A = 30◦ . If two triangle
(b) 5 points Solve the triangle which has a = 2, b =
√
3, and c = 1.
9. (a) 5 points Find the component form of the vector v which has ||v|| =
angle θ = 150◦ .
7
2
and direction
(b) 5 points Find the angle between the two vectors u = 2i and v = i +
Page 9
√
3j.
10. (a) 5 points Find the projection of of u onto v. Then write u as the sum of two
orthogonal vectors, one of which is proj v u.
√
(b) 5 points Use DeMoivre’s theorem to find 2( 3 + i)7 .
Page 10