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Sine & Cosine of
Complementary Angles

Discover why \(\sin(\theta) = \cos(90^\circ - \theta)\) and how this elegant relationship connects every pair of complementary angles in a right triangle.

Geometry

What Are Complementary Angles?

Two angles are complementary when their measures add up to exactly \(90^\circ\). In every right triangle, the two acute angles are always complementary because the three angles must sum to \(180^\circ\), and one angle is already \(90^\circ\).

C A B \(\alpha\) \(\beta\) \(90^\circ\) \(\alpha + \beta = 90^\circ\)

Reviewing \(\sin\) and \(\cos\)

In a right triangle, the sine and cosine of an acute angle are defined as ratios of sides. These ratios depend only on the angle, not the triangle's size: a property guaranteed by similarity.

\(\sin(\theta)\)
\(\displaystyle \frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos(\theta)\)
\(\displaystyle \frac{\text{adjacent}}{\text{hypotenuse}}\)

The Key Insight: Same Triangle, Two Views

When you look at a right triangle from angle \(\alpha\), the opposite side is one leg. But from the complementary angle \(\beta\), that same leg becomes the adjacent side. The hypotenuse never changes.

a (opp to A, adj to B) b (adj to A, opp to B) c (hypotenuse) A B C

The Cofunction Identity

For any acute angle \(\theta\) in a right triangle:

This is why cosine is called "co-sine": it is the sine of the complement. The prefix "co-" literally means "complement of."

Proving It with Side Ratios

a b c A B \(\sin A = \frac{a}{c}\) \(\cos A = \frac{b}{c}\) \(\sin B = \frac{b}{c}\) \(\cos B = \frac{a}{c}\) \(\sin A = \cos B\) ✓

\(\sin\) vs \(\cos\): What Changes?

From angle \(\theta\) From angle \(90^\circ - \theta\)
Opposite side \(a\) \(b\)
Adjacent side \(b\) \(a\)
Hypotenuse \(c\) \(c\)
\(\sin\) ratio \(\frac{a}{c}\) \(\frac{b}{c}\)
\(\cos\) ratio \(\frac{b}{c}\) \(\frac{a}{c}\)

Why Similarity Guarantees It

All right triangles that share the same acute angle \(\theta\) are similar (AA similarity). Similar triangles have proportional sides, so the ratio \(\frac{\text{opposite}}{\text{hypotenuse}}\) is the same regardless of the triangle's size.

\(\theta\) \(\theta\) \(\theta\) ~ ~ Same angle \(\theta\) → Same ratios

How to Apply the Identity

1
Identify the angle. Determine which acute angle \(\theta\) you are working with.
2
Find its complement. Calculate \(90^\circ - \theta\).
3
Swap the function. Replace \(\sin(\theta)\) with \(\cos(90^\circ - \theta)\), or vice versa.
4
Simplify. Use the swapped form to solve or compare expressions.

Example 1: Direct Substitution

Problem: If \(\sin(28^\circ) = 0.469\), find \(\cos(62^\circ)\).

1
Check if they are complements: \(28^\circ + 62^\circ = 90^\circ\) ✓
2
Apply the identity: \(\cos(62^\circ) = \sin(90^\circ - 62^\circ) = \sin(28^\circ)\)
3
Substitute: \(\cos(62^\circ) = 0.469\)

Example 2: Solving for an Angle

Problem: Find the value of \(x\) if \(\sin(3x + 10)^\circ = \cos(2x - 5)^\circ\).

1
Use the identity: For \(\sin A = \cos B\), we need \(A + B = 90^\circ\).
2
Set up equation: \((3x + 10) + (2x - 5) = 90\)
3
Combine: \(5x + 5 = 90\)
4
Solve: \(5x = 85\), so \(x = 17\)

Example 3: Triangle Side Ratios

Problem: In right triangle \(\triangle PQR\) with the right angle at \(Q\), \(PQ = 5\), \(QR = 12\), and \(PR = 13\). Show that \(\sin P = \cos R\).

5 12 13 P R Q
\(\sin P\)
\(\dfrac{\text{opp to }P}{\text{hyp}} = \dfrac{QR}{PR} = \dfrac{12}{13}\)
\(\cos R\)
\(\dfrac{\text{adj to }R}{\text{hyp}} = \dfrac{QR}{PR} = \dfrac{12}{13}\)

Other Cofunction Pairs

The cofunction relationship extends beyond sine and cosine. Every trig function has a "co-" partner that shares this complementary bond:

Function Cofunction Identity
\(\sin \theta\) \(\cos \theta\) \(\sin \theta = \cos(90^\circ - \theta)\)
\(\tan \theta\) \(\cot \theta\) \(\tan \theta = \cot(90^\circ - \theta)\)
\(\sec \theta\) \(\csc \theta\) \(\sec \theta = \csc(90^\circ - \theta)\)

Special Angles: Cofunction in Action

Watch the cofunction identity at work with the well-known special angles:

\(\theta\) \(\sin \theta\) \(90^\circ - \theta\) \(\cos(90^\circ - \theta)\) Equal?
\(30^\circ\) \(\frac{1}{2}\) \(60^\circ\) \(\frac{1}{2}\) ✓
\(45^\circ\) \(\frac{\sqrt{2}}{2}\) \(45^\circ\) \(\frac{\sqrt{2}}{2}\) ✓
\(60^\circ\) \(\frac{\sqrt{3}}{2}\) \(30^\circ\) \(\frac{\sqrt{3}}{2}\) ✓

Common Mistakes to Avoid

✕ Subtracting from \(180^\circ\)
Writing \(\sin(\theta) = \cos(180^\circ - \theta)\). Complementary means sum to \(90^\circ\), not \(180^\circ\). The identity uses \(90^\circ - \theta\).
✕ Swapping without changing function
Writing \(\sin(30^\circ) = \sin(60^\circ)\). You must switch from sin to cos (or vice versa) when you take the complement.
✕ Applying to non-acute angles
The cofunction identity for right-triangle trig applies to acute angles only \((0^\circ < \theta < 90^\circ)\). The right angle itself has no complement in the triangle.

Summary: Key Takeaways

1
Complementary angles sum to \(90^\circ\). In a right triangle, the two acute angles are always complementary.
2
Cofunction identity: \(\sin(\theta) = \cos(90^\circ - \theta)\) and \(\cos(\theta) = \sin(90^\circ - \theta)\).
3
Why it works: The opposite side of one angle is the adjacent side of its complement. The hypotenuse is shared.
4
Similarity guarantees that these ratios depend only on the angle, not on the triangle's size.
5
"Co-" means complement: cosine, cotangent, and cosecant all follow the same pattern.

Question 1 of 5

If \(\cos(75^\circ) = 0.259\), what is the value of \(\sin(15^\circ)\)?

Question 2 of 5

Which equation is always true for acute angle \(\theta\)?

Question 3 of 5

In a right triangle, \(\sin A = \frac{3}{5}\). Angle \(A\) and angle \(B\) are the two acute angles. What is \(\cos B\)?

Question 4 of 5

Solve for \(x\): \(\sin(2x + 14)^\circ = \cos(4x - 8)^\circ\)

Question 5 of 5

Why does the cofunction identity \(\sin \theta = \cos(90^\circ - \theta)\) work?