Complementary angle relationships — sin(90−θ) = cos θ etc.

Overview

Complementary angles are two angles that sum to 90°. The trigonometric ratios of complementary angles have a beautiful symmetry: sine and cosine swap, tangent and cotangent swap, secant and cosecant swap. These are called cofunction identities because each pair consists of a function and its "co-" partner.

The Six Cofunction Identities

$\sin(90° - \theta) = \cos \theta$ $\cos(90° - \theta) = \sin \theta$ $\tan(90° - \theta) = \cot \theta$ $\cot(90° - \theta) = \tan \theta$ $\sec(90° - \theta) = \csc \theta$ $\csc(90° - \theta) = \sec \theta$

In radians: replace90° with π/2.

Derivation from First Principles

From Right Triangle Definitions

Setup: Consider a right triangle with acute angles θ and (90° − θ).

Step 1: Label the triangle.

• Let the hypotenuse = h
• Let the side opposite to θ = a (so it's adjacent to the angle90° − θ)
• Let the side adjacent to θ = b (so it's opposite to the angle 90° − θ)

Why this matters: The same side plays different roles depending on which angle you're measuring from.

Step 2: Write sin(90° − θ) using the definition.

$\sin(90° - \theta) = \frac{\text{opposite to } (90° - \theta)}{\text{hypotenuse}} = \frac{b}{h}$

Why this step? We apply the definition: sine = opposite/hypotenuse for the angle (90° − θ).

Step 3: Recognize that b/h is cos θ.

$\cos \theta = \frac{\text{adjacent to } \theta}{\text{hypotenuse}} = \frac{b}{h}$

Why this step? Side b is adjacent to angle θ, so b/h is exactly the definition of cos θ.

Step 4: Conclude the identity.

$\sin(90° - \theta) = \frac{b}{h} = \cos \theta$

Why this works: The geometry forces the swap. The opposite side from one angle's perspective is the adjacent side from the complementary angle's perspective.

Deriving the Other Identities

By the same triangle: $\cos(90° - \theta) = \frac{\text{adjacent to } (90° - \theta)}{h} = \frac{a}{h} = \sin \theta$

The side adjacent to (90° − θ) is the side opposite to θ.

Using the sine and cosine results: $\tan(90° - \theta) = \frac{\sin(90° - \theta)}{\cos(90° - \theta)} = \frac{\cos \theta}{\sin \theta} = \cot \theta$

Why this step? Tangent is defined as sine/cosine. We substitute the identities we just proved.

$\csc(90° - \theta) = \frac{1}{\sin(90° - \theta)} = \frac{1}{\cos \theta} = \sec \theta$

$\sec(90° - \theta) = \frac{1}{\cos(90° - \theta)} = \frac{1}{\sin \theta} = \csc \theta$

$\cot(90° - \theta) = \frac{1}{\tan(90° - \theta)} = \frac{1}{\cot \theta} = \tan \theta$

Why this step? Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively. Apply reciprocals to both sides of the identities.

Worked Examples

Given: We know cos(30°) = √3/2.

Find: sin(60°) without using a calculator or table.

Solution:

Notice that 60° = 90° − 30°.

Apply the cofunction identity: $\sin(60°) = \sin(90° - 30°) = \cos(30°) = \frac{\sqrt{3}}{2}$

Why this step? We used the identity sin(90° − θ) = cos θ with θ = 30°.

Answer: sin(60°) = √3/2

Verification: You can confirm this from the standard30-60-90 triangle (sides 1: √3 : 2). For the 60° angle, opposite = √3, hypotenuse = 2, so sin(60°) = √3/2. ✓

Given: Expression tan(90° − 25°).

Simplify: Use cofunction identity.

Solution:

Apply the identity tan(90° − θ) = cot θ with θ = 25°: $\tan(90° - 25°) = \cot(25°)$

Why this step? Direct application of the tangent-cotangent cofunction pair.

We can also write this as: $\cot(25°) = \frac{1}{\tan(25°)}$

Why this step? Cotangent is the reciprocal of tangent, which may be more useful depending on context.

Answer: tan(65°) = cot(25°) = 1/tan(25°)

Given: Expression sec(90° − A) · sin(A).

Prove: It equals 1 for all valid A.

Solution:

Step 1: Apply the cofunction identity for secant. $\sec(90° - A) = \csc(A)$

Why this step? We use sec(90° − θ) = csc θ with θ = A.

Step 2: Substitute into the original expression. $\sec(90° - A) \cdot \sin(A) = \csc(A) \cdot \sin(A)$

Step 3: Use the reciprocal identity for cosecant. $\csc(A) = \frac{1}{\sin(A)}$

Why this step? Cosecant is defined as 1/sin.

Step 4: Simplify the product. $\csc(A) \cdot \sin(A) = \frac{1}{\sin(A)} \cdot \sin(A) = 1$

Why this step? Any non-zero number times its reciprocal equals 1.

Answer: sec(90° − A) · sin(A) = 1 (proven) ∎

Given: sin(3x) = cos(2x), and we need x in the range0° < x < 90°.

Find: The value of x.

Solution:

Step 1: Use the cofunction identity cos(2x) = sin(90° − 2x). $\sin(3x) = \sin(90° - 2x)$

Why this step? We convert cosine to sine using the cofunction identity so both sides have the same function.

Step 2: Set the arguments equal (for acute angles). $3x = 90° - 2x$

Why this step? If sin(α) = sin(β) and both angles are acute, then α = β (in the principal domain).

Step 3: Solve for x. $3x + 2x = 90°$ $5x = 90°$ $x = 18°$

Why this step? Basic algebra: collect like terms and divide.

Answer: x = 18°

Verification: sin(54°) = sin(90° − 36°) = cos(36°). Check: 3(18°) = 54° and 2(18°) = 36°. ✓

Common Mistakes

Why it feels right: Both "complementary" and "supplementary" sound similar and involve angle pairs.

The fix:

• Complementary angles add to 90° (right angle).
• Supplementary angles add to 180° (straight line).
• The cofunction identities only work for 90° (complementary), not 180°.
• For supplementary angles, sin(180° − θ) = sin θ, not cos θ!

Steel-man: The confusion is natural because both are standard angle-pair relationships. The mnemonic "C for Corner (90° is a corner/right angle)" helps distinguish them.

Why it feels right: The identity looks the same, and we forget that 90° ≠ π/2 when θ is in the wrong unit.

The fix:

• If θ is in degrees, use sin(90° − θ) = cos θ
• If θ is in radians, use sin(π/2 − θ) = cos θ
• Never mix units in a single expression!

Example: If θ = π/6 (radians), then sin(π/2 − π/6) = sin(π/3) = cos(π/6) = √3/2. But if you wrote sin(90° − π/6), that's nonsense (subtracting a radian measure from a degree measure).

Why it feels right: We computed the arithmetic (90 − 50 = 40) but didn't use the identity.

The fix:

• Always recognize the pattern first: "This is 90° minus something."
• Apply the cofunction identity: sin(90° − 50°) = cos(50°).
• Only then evaluate if needed.

Why this matters: In algebra problems, you often need the form cos(50°) to match other terms, not the numerical value sin(40°) ≈ 0.643.

80/20 Principle — Core to Master

The 20% that gives 80% of the value:

1. The main three pairs: sin ↔ cos, tan ↔ cot, sec ↔ csc when angles are complementary (add to 90°).
2. Quick recognition: If you see (90° − θ) or (π/2 − θ), mentally swap to the cofunction.
3. Application in equations: If sin(A) = cos(B), then A and B are complementary (A + B = 90°).

Master these three patterns, and you can solve 80% of problems involving cofunction identities.

Memory Aids

Every "co-" function pairs with its non-"co-" partner when angles are complementary.

Active Recall Practice

Now, sine and cosine are just fancy ways of comparing the sides of the triangle. Here's the cool part: when you measure sine of the30° angle, you're looking at one side compared to the longest side. But when you measure cosine of the 60° angle (the other buddy angle), you're looking at the exact same side compared to the same longest side! So sine of 30° = cosine of 60°. They're the same number!

It's like if you and your friend are looking at the same stick from opposite ends — you might call it "the left side" and your friend calls it "the right side," but it's the same stick! Sine and cosine swap roles when the angles are complementary buddies. That's why sin(90° − θ) = cos(θ) — you're just looking at the triangle from the other angle's point of view.

Connections

• Right Triangle Trigonometry — the foundation where opposite/adjacent definitions come from
• Unit Circle — cofunction identities visible as reflections across the line y = x
• Trigonometric Identities — cofunction identities are a subset of the broader identity family
• Angle Sum Formulas — can derive cofunction identities using sin(90° − θ) = sin(90°)cos(θ) − cos(90°)sin(θ)
• Even and Odd Functions — related symmetry concepts, though cofunctions show complementary symmetry
• Solving Trigonometric Equations — cofunction identities often simplify equations by unifying functions
• Graph Transformations — sin(x) shifted left by π/2 becomes cos(x), illustrating the phase relationship

#flashcards/maths

What are complementary angles? :: Two angles that sum to 90° (or π/2 radians).

Simplify: tan(90° − 40°) :: cot(40°) or 1/tan(40°) or tan(50°)

Concept Map

Hinglish (regional understanding)