Complementary Angle Identity

TipLearning Objectives
  • Recognize complementary angles in right triangles.
  • Use identities relating \(\sin\) and \(\cos\) of complementary angles.
  • Interpret why \(\sin\theta\) and \(\cos(90^\circ-\theta)\) are equal.

Key Ideas

In a right triangle, the two acute angles are complementary: \[ \theta + \phi = 90^\circ \]

Complementary trig identities:

  • \[ \sin\theta = \cos(90^\circ - \theta) \]
  • \[ \cos\theta = \sin(90^\circ - \theta) \]
  • \[ \tan\theta = \cot(90^\circ - \theta) \]

Why? Because the opposite side of one acute angle is the adjacent side of the other.

Right triangle showing one angle labeled \(\theta\) and the other \(90^\circ-\theta\), illustrating how the opposite side of one is the adjacent of the other.

Common Problem Types

Relating Sine and Cosine

Example: \(\sin 30^\circ = \cos 60^\circ\).

Solving for a Missing Ratio Using Complementarity

Given \(\sin\theta\), find \(\cos(90^\circ - \theta)\) immediately.

Interpreting Triangle Geometry

Opposite ↔︎ adjacent switch roles between acute angles.

Strategies

  • Draw a right triangle and label sides; visually see the relationship.
  • When a problem asks for \(\cos(90^\circ - \theta)\), rewrite as \(\sin\theta\).
  • Use complementary angles to avoid computing two separate trig values.

Worked Examples

Example 1

Find \(\cos(90^\circ - 25^\circ)\). \[ \cos(65^\circ) = \sin25^\circ \]

Example 2

Given \(\sin\theta = 4/5\), find \(\cos(90^\circ - \theta)\). \[ \cos(90^\circ - \theta) = \sin\theta = 4/5. \]

WarningCommon Mistakes
  • Forgetting that complementarity applies only to right triangles.
  • Mixing up \(\sin\) and \(\cos\) without using the identity.
  • Trying to compute both trig values separately.

Practice Problems

  1. Compute \(\sin(90^\circ - 40^\circ)\).
  2. If \(\cos\theta = 0.8\), find \(\sin(90^\circ - \theta)\).
  3. Are \(32^\circ\) and \(58^\circ\) complementary?
  1. = \(\cos40^\circ\)
  2. = \(0.8\)
  3. Yes, sum is \(90^\circ\).

Summary

  • Complementary angles swap opposite and adjacent.
  • \(\sin\theta = \cos(90^\circ - \theta)\) is a fundamental identity.
  • Use complementarity to simplify trig expressions.
  • When you see \(90^\circ - \theta\), think “switch sine and cosine.”
  • Visualize the triangle to avoid confusion.