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Evaluate exact circular-function values by symmetry and periodicity

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Before selecting a standard exact value, what should be done with the angle?

Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

Exact circular-function values

Reduce an angle to one full turn, locate its terminal position and measure the reference angle from the horizontal axis. Use the standard first-quadrant value, then apply the sign required by the quadrant or axis

\[\sin(\theta+2\pi)=\sin\theta,\quad \cos(\theta+2\pi)=\cos\theta,\quad \tan(\theta+\pi)=\tan\theta\]
  1. Reduce by full periods Remove complete turns before locating the terminal position, using the principal interval from 0 inclusive to 2π exclusive \(\displaystyle 0\le\theta_0<2\pi,\quad \sin(\theta+2\pi)=\sin\theta\)
  2. Measure from the horizontal axis The reference angle is the acute angle between the terminal side and the horizontal axis \(\displaystyle \alpha=\pi-\theta_0\text{ in quadrant 2},\quad \alpha=2\pi-\theta_0\text{ in quadrant 4}\)
  3. Use the standard first-quadrant value Match the reference angle with the known sine, cosine or tangent value before applying its sign \(\displaystyle \sin\left(\frac{\pi}{6}\right)=\frac{1}{2},\quad \cos\left(\frac{\pi}{3}\right)=\frac{1}{2},\quad \tan\left(\frac{\pi}{4}\right)=1\)
  4. Apply the terminal-position sign The quadrant or axis determines whether the exact value is positive, negative, zero or undefined \(\displaystyle \sin(\pi+\alpha)=-\sin\alpha,\quad \cos(2\pi-\alpha)=\cos\alpha\)

Worked example

Evaluate the exact circular-function value below

\[\sin\left(\frac{7\pi}{6}\right)\]
  1. \(\displaystyle 0\le\frac{7\pi}{6}<2\pi,\quad \theta_0=\frac{7\pi}{6}\)The angle is already in the principal interval for one full turn
  2. \(\displaystyle \alpha=\frac{7\pi}{6}-\pi=\frac{\pi}{6}\)In the third quadrant, measure the reference angle from the negative horizontal direction
  3. \(\displaystyle \sin\left(\frac{\pi}{6}\right)=\frac{1}{2}\)Use the standard first-quadrant sine value shown in this step
  4. \(\displaystyle \sin\left(\frac{7\pi}{6}\right)=-\sin\left(\frac{\pi}{6}\right)=-\frac{1}{2}\)Sine is negative in the third quadrant, so apply the negative sign to the reference value

Common mistakes

Carrying the first-quadrant sign into every quadrant
✕\(\displaystyle \sin\left(\frac{7\pi}{6}\right)=\frac{1}{2}\)✓\(\displaystyle \sin\left(\frac{7\pi}{6}\right)=-\frac{1}{2}\)
The first-quadrant value supplies the magnitude only; the terminal quadrant determines the sign of the requested function
Swapping sine and cosine exact values
✕\(\displaystyle \cos\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}\)✓\(\displaystyle \cos\left(\frac{\pi}{3}\right)=\frac{1}{2}\)
The larger value shown belongs to sine at this angle; cosine takes the smaller value
Using a half-turn as the sine period
✕\(\displaystyle \sin(\theta+\pi)=\sin\theta\)✓\(\displaystyle \sin(\theta+2\pi)=\sin\theta\)
A half-turn changes the signs of sine and cosine, while a full turn returns each value to its original value

Difficulty levels

introClassify a sign or find one standard exact value in the first turn
standardReduce an extended angle or use a reference-angle table for a basic exact value
challengeEvaluate a short exact expression or list all basic equation solutions in a multi-turn domain
examChoose a regenerated exact-form, identity or sign statement

Questions about this topic

What is a reference angle?+

For a terminal side in a quadrant, the reference angle is the acute angle it makes with the nearest horizontal axis. It is used to select a standard first-quadrant value

Why can sine and cosine change sign after a half-turn?+

Adding a half-turn moves the terminal point to the opposite side of the unit circle, changing both coordinates to their negatives. Sine and cosine therefore both change sign

Why keep an exact value instead of a decimal?+

A fraction or surd represents the value without rounding, so later calculations remain exact

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 9: Circular functions

In this chapter

Convert angle measure and use the radian arc relationLocate extended angles and generate coterminal anglesDeduce exact trigonometric ratios from one ratioUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsLocate and bisect a circular-equation rootUse local polynomial approximations for circular functions