Trigonometric Ratios of Complementary Angles

Trigonometric Ratios of Complementary Angles

TL;DR
Two angles are complementary when they add to 90°, and the trigonometric ratio of an angle equals the co-ratio of its complement — so sin(90°−θ)=cosθ, tan(90°−θ)=cotθ, and sec(90°−θ)=cscθ. This article gives all six complementary-angle identities, proves them from a right triangle, explains why the 'co-' in cosine means complement, and works through six examples — including the classic tan 1°⋅tan 2°⋯tan 89° problem.

What Are Trigonometric Ratios of Complementary Angles?

Two angles are complementary if they sum to 90°, and the trigonometric ratios of complementary angles state that each ratio of an angle equals the corresponding co-ratio of its complement. So for any acute angle θ, the angle (90°−θ) is its complement, and the two angles' ratios are linked by a fixed set of identities — also called the cofunction identities.

Here are all six, the central result of this topic:

The pattern: each ratio turns into its co-ratio (sine ↔ cosine, tangent ↔ cotangent, secant ↔ cosecant) when the angle is replaced by its complement. These are the same pairings collected in the cofunction identities; here we focus on the complementary-angle logic behind them.

Complementary is not supplementary. Complementary angles sum to 90°; supplementary angles sum to 180°. The identities above hold only for the 90° pairing — mixing the two is a common early error.

How Are the Complementary-Angle Identities Proved?

The proof needs nothing beyond a single right triangle. Take a right triangle ABC with the right angle at C. The two acute angles, at A and B, must add to 90° (the angles of any triangle sum to 180°, and C already uses 90°). So if ∠A=θ, then ∠B=90°−θ — the two acute angles are always complementary.

Now label the sides relative to ∠A=θ:

By definition, sinθ= a/c and cosθ=b/c.

Here is the key observation: the side opposite A is the side adjacent to B, and vice versa. So for the angle ∠B=90°−θ:

Therefore:

sin(90°−θ)=sinB=opposite to B/hypotenuse=b/c=cosθ

cos(90°−θ)=cosB=adjacent to B/hypotenuse=a/c=sinθ

The tangent identity follows from the quotient relation:

tan(90°−θ)=sin(90°−θ)/cos(90°−θ)=cosθ/sinθ=cotθ

and the secant/cosecant identities follow by taking reciprocals. The whole family rests on one fact: swapping the two acute angles of a right triangle swaps 'opposite' and 'adjacent.'

A Quick Numerical Check

You can verify the identities against the trigonometric ratios of specific angles you already know. Take θ=30°, so 90°−θ=60°:

Both match. The identities are not approximations; they are exact, for every angle.

Examples of Trigonometric Ratios of Complementary Angles

Example 1

Evaluate sin(18°)/cos(72°).

Notice 72°=90°−18°, so cos(72°)=sin(18°).

Thus, sin(18°)/cos(72°)=sin(18°)/sin(18°)=1.

Final answer: 1.

Example 2

Evaluate tan(26°)−cot(64°).

Test the relationship instead: 64°=90°−26°, and cot(90°−θ)=tanθ, so cot(64°)=tan(26°).

That makes the two terms identical:

tan(26°)−cot(64°)=tan(26°)−tan(26°)=0.

Final answer: 0.

Example 3

Evaluate cos(48°)−sin(42°).
Since 48°=90°−42°, we have cos(48°)=sin(42°).

Thus, cos(48°)−sin(42°)=sin(42°)−sin(42°)=0.

Final answer: 0.

Example 4

If sec(4A)=csc(A−20°), where 4A is an acute angle, find A.

Convert one side into the other's co-ratio. Since secθ=csc(90°−θ):

sec(4A)=csc(90°−4A).

So the equation becomes: csc(90°−4A)=csc(A−20°).

The cosecants are equal, so the angles are equal:

90°−4A=A−20°
110°=5A⟹A=22°.

Final answer: A=22°. (Check: 4A=88° is acute, as required.)

Example 5

Show that sin(235°)+sin(255°)=1.

Since 55°=90°−35°, we have sin(55°)=cos(35°). Substitute: sin(235°)+sin(255°)=sin(235°)+cos(235°).

By the Pythagorean identity, sin^2(35°)+cos^2(35°)=1.

Final answer: the expression equals 1.

Example 6

Evaluate tan(1°)⋅tan(2°)⋯tan(89°).

Pair each angle with its complement, noting that tan(90°−θ)=cotθ. For each pair: tan(1°)⋅tan(89°)=1, tan(2°)⋅tan(88°)=1, and so on.

The lone middle term is tan(45°)=1.

Final answer: 1.

Why Complementary-Angle Ratios Earn Their Place

These identities exist because they let you trade an awkward angle for a friendlier one and cancel terms outright — which is exactly what turns a fearsome-looking expression into a one-line answer.

Key Takeaways

Practice Before Moving On

  1. Evaluate cos(37°)/sin(53°).
  2. If tan(2A)=cot(A−18°), where 2A is acute, find A.
  3. Evaluate sec(20°)−cot(70°).

Answer to Question 1: sin(53°)=cos(37°), so the ratio is 1.
Answer to Question 2: cot(A−18°)=tan(90°−(A−18°)) gives 3A=108°, A=36°.
Answer to Question 3: cot(70°)=tan(20°), so the expression is sec(20°)−tan(20°)=1.

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