Trigonometric functions [Video Tutorial]

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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