Assigned Problems: Graphs of Trigonometric Functions
1. Sketch the graphs of the following functions on the interval [−2π, 2π]. Label all intersections with the coordinate axes.
(a) y = sin x
x intercepts: (0, 0), (±π, 0), and (±2π, 0)
y intercept: (0, 0)
(b) y = cos x
π 3π
x intercepts: ± , 0 and ± , 0
2
2
y intercept: (0, 1)
(c) y = tan x
x intercepts: (0, 0), (±π, 0) and (±2π, 0)
y intercept: (0, 0)
(d) y = cot x
1
π 3π
x intercepts: ± , 0 and ± , 0
2
2
No y intercept
(e) y = sec x
No x intercepts, y intercept: (0, 1)
(f) y = csc x
No x intercepts and no y intercept
2. For each of the following functions, determine the domain and range.
(a) y = sin x
Domain: (−∞, ∞), Range: [−1, 1]
(b) y = cos x
Domain: (−∞, ∞), Range: [−1, 1]
2
(c) y = tan x
Domain: x 6=
π
+ πk, where k is any integer; Range: (−∞, ∞)
2
(d) y = cot x
Domain: x 6= πk, where k is any integer; Range: (−∞, ∞)
(e) y = sec x
Domain: x 6=
π
+ πk, where k is any integer; Range: (−∞, −1] ∪ [1, ∞)
2
(f) y = csc x
Domain: x 6= πk, where k is any integer; Range: (−∞, −1] ∪ [1, ∞)
3. Find the amplitude and period of the following functions.
(a) y = 2 sin 5x
Amplitude: 2; Period:
2π
5
x
3
(b) y = − cos
4
2
3
; Period: 4π
4
πx
1
(c) y = − sin
2
3
1
Amplitude: ; Period: 6
2
5
4x
(d) y = cos
3
5
5π
5
Amplitude: ; Period:
3
2
Amplitude:
3
4. Sketch a graph of the following functions. Include at least two full periods. State the
amplitude and period. Label all intersections with the coordinate axes.
(a) y = 3 sin 2x
Amplitude: 3; Period: π
4
3
1
(b) y = − sin x
2
3
3
Amplitude: ; Period: 6π
2
5
(c) y = −4 cos 3x
Amplitude: 4; Period:
2π
3
6