Download 2017 ARML Trigonometry Workshop 1 Examples

2017 ARML Trigonometry Workshop
Mr. Kim and Karen Ge
1 Examples
x
E1. If sin x◦ + cos x◦ = tan x◦ , 0 < x < 135, compute b 10
c.
E2. If 6 tan−1 x + 4 tan−1 (3x) = π, compute x2 .
E3. Find the least positive real number x for which
csc x = csc 2x + csc 3x.
E4. Suppose that θ =
2π
17 .
Compute
cos θ + cos 3θ + cos 5θ + cos 7θ + cos 9θ + cos 11θ + cos 13θ + cos 15θ
1
(Mr. Kim and Karen Ge)
2 Problems
2 Problems
P1. For x and y in radians, compute the number of solutions in ordered pairs (x, y) to
the following system:
sin(x + y) = cos(x − y)
1995π 2
x2 + y 2 =
4
P2. The point A with coordinates (sin θ, cos θ) is 3 units away from the point B, which
has coordinates (2 cos 75◦ , 2 sin 75◦ ). If 0◦ ≤ θ < 360◦ , compute θ.
ANSWER TO PROBLEM 1
ANSWER TO PROBLEM 2
2
(Mr. Kim and Karen Ge)
2 Problems
P3. In triangle ABC, BC = 2. Point D is on segment AC such that AD = 1 and
CD = 2. If m∠BDC = 2m∠A, compute sin ∠A.
P4. Compute the least possible area of a non-degenerate right triangle with sides of
length sin x, cos x, and tan x, where x is a real number.
ANSWER TO PROBLEM 3
ANSWER TO PROBLEM 4
3
(Mr. Kim and Karen Ge)
2 Problems
P5. Simplify completely:
cos
2π
6π
10π
+ cos
+ cos
.
7
7
7
P6. Simplify completely:
sin 25◦ sin 35◦ sin 85◦
ANSWER TO PROBLEM 5
ANSWER TO PROBLEM 6
4
(Mr. Kim and Karen Ge)
2 Problems
P7. Find all real numbers θ with 0 ≤ θ < π, such that
(sin 3θ)(sin 3θ − cos θ) = (sin θ)(sin θ − cos 3θ)
P8. Evaluate
sin 10◦ sin 30◦ sin 50◦ sin 70◦
ANSWER TO PROBLEM 7
ANSWER TO PROBLEM 8
5
(Mr. Kim and Karen Ge)
2 Problems
P9. In 4ABC, 5 sin A + 3 cos B = 1 and 5 cos A + 3 sin B = 7. Compute sin C.
P10. (2016Chi12) Compute the number of distinct positive values of sin x such that
sin x + sin 2x + sin 3x + · · · + sin 19x = 0.
ANSWER TO PROBLEM 9
ANSWER TO PROBLEM 10
6