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SPI Mathematics · Trigonometric Ratios and Identities

Ratios of Complementary Angles

Two angles are complementary when they add to 90°, and a ratio of one equals the partner ratio of the other: sin of one is cos of the other.

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Why this matters

Six questions in the bank, four EASY and two MODERATE. Five switch a ratio to its partner using 90° minus the angle, and one is a long sum of squares that collapses when the terms are paired from the two ends.

Concept 1 of 2: Switching a ratio with 90° minus the angle

In a right triangle the two acute angles add to 90∘90^\circ. The side opposite one of them is adjacent to the other, so sin of one angle is cos of the other. The same swap pairs tan with cot and sec with cosec.

Definition

  • sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos\theta and cos⁡(90∘−θ)=sin⁡θ\cos(90^\circ - \theta) = \sin\theta.
  • tan⁡(90∘−θ)=cot⁡θ\tan(90^\circ - \theta) = \cot\theta and cot⁡(90∘−θ)=tan⁡θ\cot(90^\circ - \theta) = \tan\theta.
  • sec⁡(90∘−θ)=cosec⁡θ\sec(90^\circ - \theta) = \operatorname{cosec}\theta and cosec⁡(90∘−θ)=sec⁡θ\operatorname{cosec}(90^\circ - \theta) = \sec\theta.
  • So tan⁡θ×tan⁡(90∘−θ)=tan⁡θcot⁡θ=1\tan\theta \times \tan(90^\circ - \theta) = \tan\theta\cot\theta = 1.
  • Equations: if sin⁡x=cos⁡y\sin x = \cos y with both angles acute, then x+y=90∘x + y = 90^\circ.

Complementary angle rules

sin⁡(90∘−θ)=cos⁡θ,tan⁡(90∘−θ)=cot⁡θ,sec⁡(90∘−θ)=cosec⁡θ\sin(90^\circ - \theta) = \cos\theta,\quad \tan(90^\circ - \theta) = \cot\theta,\quad \sec(90^\circ - \theta) = \operatorname{cosec}\theta

Worked example

Evaluate sin⁡25∘cos⁡65∘+tan⁡20∘tan⁡70∘\dfrac{\sin 25^\circ}{\cos 65^\circ} + \tan 20^\circ \tan 70^\circ.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Trigonometric Ratios and Identities · Ratios of Complementary Angles

The value of 2tan⁡53cot⁡37−cot⁡80tan⁡10=\dfrac{2\tan 53}{\cot 37} - \dfrac{\cot 80}{\tan 10} =

Keeping the same ratio

sin⁡(90∘−A)\sin(90^\circ - A) is cos⁡A\cos A, not sin⁡A\sin A. The ratio always changes to its partner when the angle becomes 90∘90^\circ minus it.

Switching when the angles do not add to 90°

sin⁡40∘\sin 40^\circ equals cos⁡50∘\cos 50^\circ, not cos⁡40∘\cos 40^\circ. Check that the two angles add to 90∘90^\circ before you swap the ratio.
Drill 4 more on switching a ratio with 90° minus the angle

Concept 2 of 2: Pairing terms of a long sum or product

A long sum like sin⁡25∘+sin⁡210∘+…\sin^2 5^\circ + \sin^2 10^\circ + \dots looks impossible, but the first and last terms are complementary. Pair them from the outside in: each pair of squares adds to 1, and each pair of tangents multiplies to 1.

Definition

  • sin⁡2θ+sin⁡2(90∘−θ)=sin⁡2θ+cos⁡2θ=1\sin^2\theta + \sin^2(90^\circ - \theta) = \sin^2\theta + \cos^2\theta = 1. The same holds for cos⁡2\cos^2.
  • tan⁡θ×tan⁡(90∘−θ)=1\tan\theta \times \tan(90^\circ - \theta) = 1.
  • Count the terms first: from aa to bb in steps of dd there are b−ad+1\frac{b - a}{d} + 1 terms.
  • Add back any term with no partner: sin⁡245∘=12\sin^2 45^\circ = \frac{1}{2}, sin⁡290∘=1\sin^2 90^\circ = 1, tan⁡45∘=1\tan 45^\circ = 1.

Number of terms in a list of angles

terms=last−firststep+1\text{terms} = \frac{\text{last} - \text{first}}{\text{step}} + 1

Worked example

Find cos⁡210∘+cos⁡220∘+cos⁡230∘+⋯+cos⁡280∘\cos^2 10^\circ + \cos^2 20^\circ + \cos^2 30^\circ + \dots + \cos^2 80^\circ.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Trigonometric Ratios and Identities · Ratios of Complementary Angles

The value of sin⁡25∘+sin⁡210∘+sin⁡215∘+⋯+sin⁡290∘\sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + \dots + \sin^2 90^\circ is equal to

Forgetting the unpaired terms

In a sum of sin⁡2\sin^2 running up to 90∘90^\circ, the 45∘45^\circ term pairs with itself and the 90∘90^\circ term has no partner. Add 12\frac{1}{2} and 1 for them after counting the pairs.

Miscounting the terms

From 10∘10^\circ to 80∘80^\circ in steps of 10∘10^\circ there are 8 terms, not 7. Use last minus first, divided by the step, plus 1.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Switching a ratio with 90° minus the angle

    Complementary angle rules

    sin⁡(90∘−θ)=cos⁡θ,tan⁡(90∘−θ)=cot⁡θ,sec⁡(90∘−θ)=cosec⁡θ\sin(90^\circ - \theta) = \cos\theta,\quad \tan(90^\circ - \theta) = \cot\theta,\quad \sec(90^\circ - \theta) = \operatorname{cosec}\theta
  • Pairing terms of a long sum or product

    Number of terms in a list of angles

    terms=last−firststep+1\text{terms} = \frac{\text{last} - \text{first}}{\text{step}} + 1

Watch out for (4)

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