Download Problems from Trigonometry

Math 280
Fall 2012
Problems from Trigonometry
1. Evaluate the trigonometric functions, do not use a calculator.
(a) cos(2π/3)
(b) tan(15π/4)
(c) sec(7π/6)
(d) cos(π/12)
(e) tan(3π/8)
2. Prove the trigonometric identity
1 − cos θ
sin θ
(b) sec(π/2 − θ) = csc θ
(a) tan 2θ =
(c) sec(x + π) = − sec x
3. Solve the equations below. Find all solutions in the interval [0, 2π]
(a) tan θ = 1
(b) 2θ cos θ + θ = 0
(c) cos 3x = sin 3x
(d) cos x = 2
4. Evaluate without using a calculator.
√
(a) arcsin( 3/2)
(b) cos(arccos(−1))
(c) arccos(cos(7π/6))
√
(d) arctan( 3)
(e) arctan(tan π/4)
(f) arctan(tan(3π/4))
(g) tan(arctan(1))
5. Verify the identities
(a) arcsin x + arcsin(−x) = 0
(b) arccos x + arccos(−x) = π
6. Simplify the expressions (draw a right triangle).
(a) cos(arcsin(x))
(b) sin(arccos(x/2))
(c) cos(2 arcsin x)
(d) cos(arctan x)
(e) cot(arctan 2x)
7. (a) Given that sin θ = −4/5, π < θ < 3π/2, evaluate the remaining
five trigonometric functions.
(b) Given that sec θ = 13/5, 0 < θ < π/2, evaluate the remaining five
trigonometric functions.
√
8. Express y = sin x − 3 cos x in the form y = A sin(x + b).
9. Find a formula for the curve graphed below.
10. (a) Design a sine (or cosine) function that as a period of 12 hours, a
minimum value of −4 at t = 0 hours and a maximum value of 6
at t = 6 hours.
(b) Design a sine (or cosine) function that as a period of 24 hours, a
minimum value of 10 at t = 3 hours and a maximum value of 16
at t = 15 hours.
In both problems begin with a sine or cosine function and then use
translations (shifts) and scaling (compressions, stretches) to construct
the function.
11. The sides of a triangle are 5, 10 and 12 units long. What are the angles
in this triangle?
12. In a triangle one side has length 10 units, and the adjacent angles are
20 and 120 degrees respectively. How long are the remaining sides?
13. A point P is located 200 ft from the base of a tree, the line connecting
P to the top T of the tree makes an angle of 22 degrees with the
horizontal. How tall is the tree?
sin θ
= 1, compute
θ→0 θ
14. Given that lim
sin(aθ)
bθ
1 − cos θ
(b) lim
θ→0
θ
tan θ
(c) lim
θ→0
θ
(a) lim
θ→0
d
d
15. Given that dx
sin x = cos x and dx
cos x = − sin x , use the
quotient rule to establish formulas for the derivatives of tan x and sec x.
16. Compute
Z π/2
sin2 t dt.
0
cos t
, y = 3 + sin 2t − 2 sin2 t
2 + sin t
where 0 ≤ t ≤ 2π. How high does it fly? How far left and right of the
origin does it fly?
17. The path of a fly is given by x =
18. A pole of length L is carried horizontally around a corner where a
3-ft-wide hallway meets a 4-ft-wide hallway at a right angle.
(a) Find the relationship between L and the angle θ when the pole
simultaneously touches both walls and the corner.
(b) What is the longest pole that can be carried around the corner?
19. The hour hand of a clock is 5 cm long, the minute hand is 4 cm long.
If the coordinate origin is located at the center of the clock, find the
coordinates of the tip of the minute hand and the tip of the hour hand
at 10:00.
"
20. Let A =
cos θ − sin θ
sin θ cos θ
#
.
(a) What is the determinant of A?
(b) Compute (and simplify) A2 , and find det A2 .
(c) Compute (and simplify) An , and find det An .