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Trigonometric functions

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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