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Use exact trigonometric identities and quadrant information

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Which is the Pythagorean identity?

Year 12, VCE Units 3 & 4, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

Exact trigonometric relationships

Use the Pythagorean identity to find a missing squared ratio, then choose the square-root sign from the quadrant. Use the sine-over-cosine quotient only after the relevant sine and cosine values are known

\[\sin^2\theta+\cos^2\theta=1,\quad\tan\theta=\frac{\sin\theta}{\cos\theta}\]
  1. Keep both powers The Pythagorean identity applies to the squares of sine and cosine \(\displaystyle \sin^2\theta+\cos^2\theta=1\)
  2. Find the missing square Substitute the known ratio and rearrange before taking a square root \(\displaystyle \sin\theta=\frac{3}{5}\Rightarrow\cos^2\theta=1-\frac{9}{25}=\frac{16}{25}\)
  3. Use the quadrant sign Choose the sign of the square root from the quadrant containing the angle \(\displaystyle \theta\in\mathrm{QII}\Rightarrow\cos\theta=-\sqrt{\frac{16}{25}}=-\frac{4}{5}\)
  4. Use the quotient in order Tangent is sine divided by cosine \(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{3/5}{-4/5}=-\frac{3}{4}\)

Worked example

For the given cosine and quadrant, find sine and tangent exactly

\[\cos\theta=-\frac{5}{13},\quad\theta\in\mathrm{QIII}\]
  1. \(\displaystyle \sin^2\theta=1-\left(-\frac{5}{13}\right)^2=1-\frac{25}{169}=\frac{144}{169}\)Rearrange the Pythagorean identity and substitute the known cosine
  2. \(\displaystyle \sin\theta=-\sqrt{\frac{144}{169}}=-\frac{12}{13}\)Sine is negative in QIII, so choose the negative square root
  3. \(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-12/13}{-5/13}=\frac{12}{5}\)Divide sine by cosine; the two negative signs give a positive tangent

Common mistakes

Dropping the powers
✕\(\displaystyle \sin\theta+\cos\theta=1\)✓\(\displaystyle \sin^2\theta+\cos^2\theta=1\)
The Pythagorean identity applies to the squares of sine and cosine, not their unsquared sum
Ignoring the quadrant sign
✕\(\displaystyle \theta\in\mathrm{QII},\quad\cos\theta=\sqrt{\frac{16}{25}}=\frac{4}{5}\)✓\(\displaystyle \theta\in\mathrm{QII},\quad\cos\theta=-\sqrt{\frac{16}{25}}=-\frac{4}{5}\)
Cosine is negative in QII, so the negative square-root branch is required
Inverting the tangent quotient
✕\(\displaystyle \tan\theta=\frac{\cos\theta}{\sin\theta}\)✓\(\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}\)
The quotient identity defines tangent as sine divided by cosine

Difficulty levels

introUse the Pythagorean identity to find one missing exact ratio
standardDeduce related ratios using a given quadrant or complementary-angle relationship
challengeUse a quotient, verify an identity or combine several exact relationships
examApply identities, exact ratios and quadrant information in one problem

Questions about this topic

Why is a square root sign needed after using the identity?+

The identity gives a squared ratio, so both positive and negative roots are possible until the quadrant determines the sign

How is the sign of a trigonometric ratio chosen?+

Use the quadrant: sine is positive above the horizontal axis, cosine is positive to the right, and tangent has the sign of their quotient

What is the quotient identity for tangent?+

Tangent is sine divided by cosine, provided cosine is non-zero

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations

In this chapter

Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems