Summary — What This Chapter Covers
- Sketch y = sin x, cos x and tan x (degrees or radians) and simple transformations such as y = 3 sin x or y = 1 − cos 2x.
- Exact values for 30°, 45°, 60° and related angles.
- sin−1, cos−1 and tan−1 give principal values only.
- Identities: tan θ ≡ sin θ / cos θ and sin2θ + cos2θ ≡ 1.
- Find all solutions of a trig equation in a given interval.
1. Graphs and Exact Values
Graphs
Period, amplitude and exact values
sin and cos have period 360° (2π) and range −1 to 1; tan has period 180° (π) and vertical asymptotes at 90°, 270°, …. For y = a sin bx the amplitude is |a| and the period is 360°/b. Related angles come from symmetry: sin(180° − θ) = sin θ, cos(360° − θ) = cos θ, tan(180° + θ) = tan θ.
Exact values
sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
sin 45° = cos 45° = 1/√2, tan 45° = 1
sin 60° = √3/2, cos 60° = ½, tan 60° = √3
Worked Example
Find the exact values of cos 150° and sin(4π/3).
Practice Question
State the range and the period of y = 1 − cos 2x.
2. Inverse Functions and Identities
Identities
Principal values and the two P1 identities
Because sin, cos and tan are not one-one, their inverses are defined on restricted domains. The calculator value is the principal value:
Principal values
sin−1 x: −π/2 ≤ y ≤ π/2 · cos−1 x: 0 ≤ y ≤ π · tan−1 x: −π/2 < y < π/2
Identities
tan θ ≡ sin θ / cos θ sin2θ + cos2θ ≡ 1
To prove an identity, start from one side (usually the more complicated) and work to the other. Do not treat it as an equation and do the same thing to both sides.
Worked Example
Prove that 1/cos θ − cos θ ≡ sin θ tan θ.
Practice Question
Find the exact values of sin−1(−½) and cos−1(−½).
3. Solving Trigonometric Equations
Equations
Every solution in the interval
2
2
2
Worked Example
Solve 3 sin2θ − 5 cos θ − 1 = 0 for 0° ≤ θ ≤ 360°.
Practice Question 1
Solve 2 sin 2x = 1 for 0° ≤ x ≤ 180°.
Practice Question 2
Solve sin θ = 2 cos θ for 0° ≤ θ ≤ 360°.
Memorise
sin symmetry
sin(180° − θ) = sin θ
cos symmetry
cos(360° − θ) = cos θ
tan period
tan(θ + 180°) = tan θ