Benchmark Angles

TipLearning Objectives
  • Recall exact trig values for key angles.
  • Use special right triangles to generate benchmark values.
  • Apply benchmark angles to evaluate trig expressions quickly.

Key Ideas

Benchmark angles commonly used in trig:

Angle Radians Exact Values (sin, cos, tan)
\(30^\circ\) \(\pi/6\) \(\sin=\tfrac12\), \(\cos=\tfrac{\sqrt3}{2}\), \(\tan=\tfrac{1}{\sqrt3}\)
\(45^\circ\) \(\pi/4\) \(\sin=\tfrac{\sqrt2}{2}\), \(\cos=\tfrac{\sqrt2}{2}\), \(\tan=1\)
\(60^\circ\) \(\pi/3\) \(\sin=\tfrac{\sqrt3}{2}\), \(\cos=\tfrac12\), \(\tan=\sqrt3\)
\(90^\circ\) \(\pi/2\) \(\sin=1\), \(\cos=0\), undefined tan

Derived from special triangles:

  • 30–60–90 triangle
  • 45–45–90 triangle

Special right triangles showing the 30–60–90 and 45–45–90 ratio relationships.

Common Problem Types

Evaluating Sin/Cos/Tan Exactly

Example: \(\sin 30^\circ = 1/2\).

Using Special Triangles Instead of Calculators

Helps with exact radical values.

Recognizing Undefined Values

Example: \(\tan 90^\circ\) undefined.

Relating Complementary Angles

Use \(\sin 30^\circ = \cos 60^\circ\), etc.

Strategies

  • Memorize the two special triangles.
  • Redraw triangles to re-derive values if forgotten.
  • Simplify radicals fully.

Worked Examples

Example 1

Find \(\cos 60^\circ\):
\[ \cos 60^\circ = 1/2 \]

Example 2

Evaluate \(\tan 45^\circ\):
\[ 1 \]

WarningCommon Mistakes
  • Mixing up 30° and 60° values.
  • Forgetting radicals in exact forms.
  • Using calculator when exact value is expected.

Practice Problems

  1. Compute \(\sin 45^\circ\).
  2. Find \(\tan 30^\circ\) (exact).
  3. Evaluate \(\cos \frac{\pi}{4}\).
  1. \(\sqrt2/2\)
  2. \(1/\sqrt3\)
  3. \(\sqrt2/2\)

Summary

  • Benchmark angles come from special right triangles.
  • Know exact values for 30°, 45°, 60°, 90°.
  • Use radicals, not decimals.
  • Rebuild special triangles quickly to recall ratios.
  • Complementary angles swap sine and cosine.