Use exact trigonometric identities and quadrant information

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Coordinates determine signs

On the unit circle, cosine gives the horizontal coordinate and sine gives the vertical coordinate. The point is one unit from the origin, giving the identity shown. Taking a square root gives a magnitude; the quadrant selects its sign

\cos^2\theta+\sin^2\theta=1

Year 11, VCE Units 1 & 2, Chapter 7, 1 difficulty level, No calculatorBuild a worksheet

Unit-circle foundation

Separate magnitude from sign

Identity sin²θ+cos²θ=1 for every real angle
Quotient tan θ=sin θ/cos θ only when cos θ≠0

Worked example

Given cos B=−5/13 in quadrant III, find sin B and tan B

\sin^2B=1-\frac{25}{169}=\frac{144}{169}

\(\displaystyle \sin B=-\frac{12}{13},\qquad\tan B=\frac{12}{5}\)Sine is negative and tangent positive in quadrant III

Common mistakes

Square-root sign

✕sin B=12/13 automatically✓use the quadrant after finding the magnitude

Squaring loses the sign

Difficulty levels

introRecover a missing signed ratio

Questions about this topic

Does sin²θ mean sin(θ²)?+

No. It means (sin θ)²

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 7: Trigonometry and transformations

In this chapter

Convert angle measure and use the radian arc relation Area of a sector Use the sine rule in a non-right-angled triangle Use the cosine rule in a non-right-angled triangle Trigonometry in three dimensions Exact circular-segment area and perimeter Choose perpendicular sections and calculate spatial angles Use exact trigonometric identities and quadrant information Compound, double-angle and reciprocal identities Transform and reconstruct sine and cosine functions Combine waves and convert trigonometric sums and products Map or recover points and finite coordinate sets Compose two plane transformations in order Rotate, reflect and interpret fixed points and area