This tutorial covers material encountered in chapter 6 of the VCE Mathematical Methods Textbook, namely:
- Radians
- The sine, cosine and tangent functions
- The unit circle and its properties
- Trigonometric identities
- Graphs of trigonometric functions
- General solutions of trigonometric equations
Q1 – Solving Sine & Cosine Functions and Unit Circle
Q2 – Graphs of Sine & Cosine Functions
Q3 – Solving Tangent Functions and Unit Circle
Q4 – Solving Trigonometric Functions
Q5 – Intercepts of Trigonometric Functions
Q6 – General Solutions of Trigonometric Equations
Worksheet
Q1. Solve the following equations for x \in [0,2\pi]:
(a) \sin(x)=\frac{1}{2}
(b) 2\cos(x)=-1
(c) \sqrt{2}\sin(x)+1=0
(d) \cos(2x)=\frac{-1}{\sqrt{2}}
(e) 2\sin(3x)-1=0
Q2. Sketch the graph of each of the following trigonometric functions, showing one cycle. Find the period and amplitude for each and label any axis intercepts:
(a) f(x)=\sin(3x)
(b) f(x)=\cos(\pi x)
(c) g(x)=-3\sin(x)
(d) g(x)=2\cos(3x)+1
Q3. Solve each of the following equations for x \in [-\pi,\pi]
(a) \tan(x)=\sqrt{3}
(b) \tan(x)=1
(c) \tan(2x)=-1
Q4. Solve the equation \sin(x)=\sqrt{3}\cos(x) for x\in [0,2\pi]
Q5. The graphs of f(x)=\cos(x) and g(x)=a\sin(x) for a\in \R intersect at x=\frac{\pi}{4}
(a) Find a
(b) If x\in [0,2\pi] find any other points of intersection
Q6. Find the general solution to the following equations:
(a) \sin(3x)=1
(b) \cos(2x)=0
(c) \tan(x)=-1
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