Chapter 8: Trigonometric Functions

Question 1

Describe the features of a periodic graph

Answer 1

• Period: length of one wave cycle
• Principal axis: horizontal line which the wave oscillates about
• Maximum and minimum point: highest point of the wave’s crest and the lowest point of the wave’s trough respectively
• Amplitude: distance between the wave’s maximum/minimum point and the principal axis

Formulas:
Principal axis = (max+min)/2
Amplitude = (max-min)/2

Question 2

State the features of, draw the graph of and state the general functions of y=sinx and y=cosx

Answer 2

sin(x):
- Period: 2π
- Principal axis: 0
- Maximum point: (π/2,1)
- Minimum point: (3π/2, -1)
- x-intercepts: (0,0), (π,0) and (2π,0)
- Amplitude: 1

cos(x):
- Period: 2π
- Principal axis: 0
- Maximum point: (0,1) and (2π , 1)
- Minimum point: (π, -1)
- x-intercepts: (π/2,0) and (3π/2,0)
- Amplitude: 1

sin(x) is a horizontal translation of cos(x) by π/2 units to the right:
sin(x) = cos (x-π/2)
cos(x) = sin(x+π/2)

General sine & cosine function:
y = a sinb(x-c)+d
y = a cosb(x-c)+d

|a|: amplitude [vertical stretch by a units + reflection in x-axis if a<0]
2π/b = period [horizontal stretch by 1/b units]
c: horizontal translation [by c units]
d: vertical translation [by d units] and principal axis

Question 3

Draw the graph for and state the features of y=tanx

Answer 3

tan(x):
- Period: π
- Principal axis = 0
- Amplitude: undefined
- Vertical asymptotes:
x=π/2 and x=3π/2
- x-intercept: (0,0), (π,0) and (2π,0)