Trig. Review, Part 1
1. Find the reference number for the following. If the reference number is a
value whose terminal point you know, also find the terminal point.
(1) −11π
3
(2) 3.5
(3) 7π
3
(4) 7π
3 + 1001π
(5) −8.6
2.
• A point (x, − 27 ) is on the unit circle in the fourth quadrant. Find x.
• Suppose sin(x) = −0.4 and cos(x) > 0. Find cos(x), tan(x), sec(x), csc(x), cot(x).
• Why is the identity sin2 (x) + cos2 (x) = 1 always true? Given that, prove
the other two Pythagorean identities.
• Write tan(x) in terms of csc(x) if tan(x) < 0 and sec(x) < 0
• Suppose sin(t) = −2/5 and sec(t) < 0. Find the terminal point for t on the
unit circle. Also, find sin(2π − t) + cos(t + 3π) and tan(t − 7pi) (you should
not need to use any addition formulas).
) cot( 5π
cos( −23π
• Without using a calculator, evaluate sin( 11π
4 ) +√
6 )
√ 3
2
• Without using a calculator, evaluate ( 3sec(200)) − ( 3sec(200))2
3a. Consider the following graph, where c =
π
2
and d = 7. Write this as a
as sin k(x − b) and ac cos k(x − b) for a, k > 0.
3b. Consider the following graph, where c = 2 and d = 42. Write this as a
as sec k(x − b) and ac csc k(x − b) for a, k, b > 0.
3c. Sketch the graphs of −5 cot(2x + π) and 2 tan(x + 1)
1
Trig. Review, Part 1
4. Simplify:
(1) (f ◦g)(t) where f (x) =
q
4−
4
x2 (
√
√
x + 1 x − 1) and g(t) = sec(t). Assume
7π
2
< t < 4π.
9π
(2) sin(x + 9π
6 ) and cos(x + 6 ) if csc(x) = −3/2 and the terminal point for x
is in the third quadrant.
2
3/2
and g(t) = 75 csc(t) and 0 < t < π2
(3) (f ◦ g)(t) where f (x) = (25x −49)
x
(4) csc(t) + sec(t) for terminal point of t in quadrant III
(5) Write sec4 (x) as an expression with no powers of trig. functions.
5. Find all solutions for the following equations:
(1) 2 cos2 (x) + cos(x) = 1
(2) x = cos(t + 5π
csc(t) = −4 and tan(t) > 0
4 ) where
√
(3) 2 cos(t) tan(t) = − 3 tan(t)
(4) sin(2x) sin(x) = cos(x)
(5) cos(5x) = √12
(6) cos(x) cos(3x) − sin(x) sin(3x) = 0.5
(7) cos(2x) + cos(x) = 2
(8) 2 cos2 (x) + sin(x) = 1
2