Download Math 50 Exercises 1. Change from degree to radian

Math 50 Exercises
1. Change from degree to radian:
a. 125◦
b. −32◦
c. 450◦
d. −865◦
2. Change from radian to degree:
π
a.
6
3π
b. −
5
π
c.
2
4π
d. −
7
23π
e. −
6
20
f.
7
g. −1
h. 3.14
3. Find the exact value of the following:
3π
a sin
4
!
π
b. cos −
6
5π
c. sec
4
!
7π
d. tan −
3
!
π
e. csc −
2
21π
f. sec
4
1
4. Suppose θ is an angle in standard position and P is the terminal
point of θ, find the value of the six trigonometric functions of θ:
a. P = (1, −3)
√ 

1
2
b. P =  ,
3 3
 √

√
3
c. P = − , − 3
4
√ d. P = −2, 2
5. Find the value of the other five trigonometric functions of θ from
the information given:
2
a. cos θ = , terminal point of θ is in first quadrant.
5
√
3
b. sin θ =
, terminal point of θ is in second quadrant.
4
1
c. tan θ = − , terminal point of θ is in second quadrant.
5
7
d. csc θ = − , terminal point of θ is in fourth quadrant.
2
e. sec θ = 2, terminal point of θ is in fourth quadrant.
√
− 3
f. cot θ =
, terminal point of θ is in second quadrant.
5
1
g. cos θ = − √ , terminal point of θ is in second quadrant.
3
√
5
h. cos θ =
, terminal point of θ is in fourth quadrant.
6
2
6. If sin(θ) = , sin(−θ) =?
3
1
7. If cos(θ) = , cos(−θ) =?
4
√
8. If tan(θ) = 2, tan(−θ) =?
1
9. If sin(θ) = , sin(4π − θ) =?
3
4
10. If cos(θ) = − , cos(−θ − 6π) =?
5
2
11. Let f (x) = 3 sin(2x − 1).
a. Find the amplitude of f .
b. Find the period of f .
c. Graph f . In your graph, for at least one cycle of f , indicate the
coordinates of the x intercept, the maximum and minimum values of
f.
π
12. Let f (x) = − cos(x + ).
3
a. Find the amplitude of f .
b. Find the period of f .
c. Graph f . In your graph, for at least one cycle of f , indicate the
coordinates of the x intercept, and the maximum and minimum values
of f .
13. Let f (x) = tan(x − 4).
a. Find the period of f .
b. Graph f . In your graph, for at least one cycle of f , indicate the
coordinates of the x intercept, and the location of the vertical asymptotes.
14. Let f (x) = 2 sec(πx + 1).
a. Find the period of f .
b. Graph f . In your graph, for at least one cycle of f , indicate the
coordinates of the x maximum and minimum values of f ,
and the location of the vertical asymptotes.
3
15. Verify the Identity:
cos2 x − sin2 x
a.
= cot x − tan x
cos x sin x
cos x − sin x
= csc x − sec x
b.
cos x sin x
tan x + 1
c.
= sin x + cos x
sec x
cos x
sin x + 1
=
d.
cos x
1 − sin x
e. tan x + cot x = sec x csc x


!
(cos
h)
−
1
sin(x + h) − sin x
sin h


= (sin x)
+ (cos x)
f.
h
h
h
cos x − cos 3x
= tan x
g.
sin x + sin 3x
sin x
h.
= (csc x)(1 + cos x)
1 − cos x
sin 2x
= 1 − cos 2x
i.
cot x
sin(x − y) tan x − tan y
j.
=
sin(x + y) tan x + tan y
2 − sec2 x
= cos 2x
sec2 x
sin x + sin 5x
l.
= tan 3x
cos x + cos 5x
sin x + sin y
x−y
m.
= − cot
cos x − cos y
2
cos x − cos y
x+y
x−y
= − tan
tan
n.
cos x + cos y
2
2
k.
4
16. Find exact value of the following:
a. sin−1 (0)
√ 
3
b. sin−1  
2
c. cos−1 (−1)
 √ 
2
d. cos−1 − 
2
e. tan−1 (−1)
!
1
f. arcsin
2
 √ 
2
g. arccos − 
2
17. Find exact value:
a. arcsin(sin(3π))
9π
b. arccos(cos(− ))
2
−21π
))
c. tan−1 (tan(
4
d. sin−1 (sin(4.579))
e. cos−1 (cos(−13))
18. Find the exact value:
1
a. sin(arccos( ))
3
2
b. cos(arcsin(− ))
5
−1
c. sec(tan (3))
2
d. tan(cos−1 ( ))
7
3
e. csc(sin−1 ( √ ))
13
5
19. Rewrite the following in terms of x so that the expression is free
of any trigonometric functions:
a. sin(arccos x)
b. cos(tan−1 x)
c. tan(sin−1 x)
20. Solve the given equation. If the range for x is indicated, only solve
for x over that interval. Otherwise, solve for all possible solutions.
√
2
a. cos x =
2
√
3
b. sin x = −
2
1
c. sin x = −
2
√
d. tan x = 3
e. cos x = −1
f. sin x = 0
g. 2 sin x + 1 = 0, 0 ≤ x ≤ 2π
√
h. 2 cos x − 3 = 0, 0 ≤ x ≤ 2π
i. cos x = cot x, 0 ≤ x ≤ 2π
j. tan x = −2 sin x, 0 ≤ x ≤ 2π
k. 2 cos2 x + 3 sin x = 0, 0 ≤ x ≤ 2π
l. cos 2x = sin x
m. sin 2x = 1 + cos 2x
n. cos 2x = cos x − 1
6
21. Solve the triangle. If the triangle is impossible, explain. If there is
more than one possible triangle, find both.
a. A = 115◦ , B = 40◦ , c = 7
b. A = 22◦ , b = 5, c = 3
c. A = 92◦ , C = 41◦ , c = 10
d. a = 8, b = 5, c = 10
e. A = 50◦ , a = 10, b = 12
f. A = 46◦ , B = 76◦ , b = 5
7
22. If v = h3, −5i and u = h2, 4i are vectors in R2 , find the following:
a. 3v + 4u
b. 5u − 2v
c. |v|
d. Find a unit vector that is parallel to u
e. v · u
f. Is v and u perpendicular to each other? Explain
g. Find the angle formed between v and u
8
23. The following points are in rectangular coordinate, change them
to polar coordinate. Use exact value whenever possible.
a. (0, 1)
b. (3, 3)
√
c. (− 3, 1)
d. (0, −1)
√
e. (−1, − 3)
f. (−2, −3)
g. (2, −5)
h. (4, −1)
24. The following points are in polar coordinate, change them to rectangular coordinate. Use exact value wheneven possible.
a. (1, 0)
b. (−1, 0)
!
2π
c. 2,
3
!
5π
d. −3, −
6
!
9π
e. −1,
4
!
π
f. 1, −
2
g. (−1, 1)
h. (−4, −2)
25. Describe the curve given by the following polar equation:
a. r = 10
π
b. θ = −
3
c. r = 2θ, 0 ≤ θ < ∞
d. r = 10 sin θ
e. r = 8 cos θ
f. r = −2 sin θ
9