1
Some commonly occurring angles
θ
0
sin θ
0
cos θ
1
tan θ
0
π
6
1
2
√
2
2
√
3
2
√
3
2
√
2
2
√
1
2
√
1
0
undefined
π
4
π
3
π
2
2
3
3
1
3
Arcs in the unit circle
If we start with an arc θ in the first quadrant, then in the counterclockwise direction the arc
symmetric with respect to the sine axis is π − θ, the arc symmetric to that with respect to the
cosine axis is π + θ, and finally the arc symmetric to that with respect to the sine axis is 2π − θ.
Going in the counterclockwise direction the arc symmetric to θ with respect to the cosine axis is
−θ, the arc symmetric to that with respect to the sine axis is θ − π, and the arc symmetric to that
with respect to the cosine axis is −π − θ.
Points in the circle that are in the same horizontal line have the same sine and opposite cosines;
points that are on the same vertical line have the same cosine and opposite sines. Also antidiametrical points on the circle (i.e. θ and π + θ) have the same tangent and cotangent but opposite sines
and cosines.
sin
π−θ
b
θ
b
−π − θ
cos
π+θ
b
θ−π
−θ
b
2π − θ
sin(π − θ) = sin(θ)
cos(π − θ) = − cos(θ)
sin(−π − θ) = sin(θ)
cos(−π − θ) = − cos(θ)
sin(π + θ) = − sin(θ)
cos(π + θ) = − cos(θ)
sin(θ − π) = − sin(θ)
cos(θ − π) = − cos(θ)
sin(2π − θ) = − sin(θ)
cos(2π − θ) = cos(θ)
sin(−θ) = − sin(θ)
cos(−θ) = cos(θ)
3
Exercises
1. Calculate the following. Give exact answers whenever possible.
−1 1
(a) sin
2
√ !
2
(b) cos−1 −
2
√ !!
3 11
−1
(c) cos sin
10
(d) tan cos−1 (−0.4)
23π
(e) cos−1 cos
6
5π
(f) sin−1 sin
4
2π
(g) sin−1 sin
3
5π
(h) cos−1 cos
6
2. Give all solutions of the following equations in the interval [0, 2π).
√
3
(a) sin x = −
2
√
2
(b) cos x =
2
1
(c) sin x =
2
1
(d) cos x = −
2
1
(e) sin x =
3
(f) 4 cos2 x − 3 = 0
(g) 2 sin2 x + sin x = 0
(h) 2 sin2 x + sin x − 1 = 0
(i) 4 cos3 x = cos x
(j) 9 sin4 x − 8 sin2 x = 1
3. Find all solutions of the equations of the previous exercise in the interval (−π, π].
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