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Trigonometric Values for Standard Angles: Easy Table

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The standard angles 0°, 30°, 45°, 60° and 90° have exact trigonometric values built from small whole numbers and square roots, and they appear in almost every trigonometry problem. Their sines are 0, 1/2, √2/2, √3/2 and 1; their cosines run the same list backwards.

These values matter because examiners and textbooks use them constantly to test whether you understand trigonometry rather than just punching a calculator. Knowing them by heart turns long evaluations into quick mental arithmetic.

Key Concepts

  • Standard angles: 0°, 30°, 45°, 60°, 90°, equivalently 0, π/6, π/4, π/3, π/2 radians.
  • The √n/2 pattern: sin 0°, 30°, 45°, 60°, 90° equal √0/2, √1/2, √2/2, √3/2, √4/2. One pattern memorizes the whole sine row.
  • Cosine mirrors sine: cos θ = sin(90° − θ), so the cosine row is the sine row reversed.
  • Source triangles: 30° and 60° come from halving an equilateral triangle; 45° comes from an isosceles right triangle.
  • tan 90° is undefined: cos 90° = 0, and division by zero is not allowed.

Formulas

  • Sine row: sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
  • Cosine row: cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
  • Tangent row: tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° is undefined.
  • Derivation: in a 30°-60°-90° triangle with hypotenuse 2, the sides are 1, √3, 2; in a 45°-45°-90° triangle with legs 1, the hypotenuse is √2.

Worked Example

Evaluate 2 sin 30° + 3 cos 60° − tan 45°.

  1. Substitute the standard values: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1.
  2. 2 sin 30° = 2 × (1/2) = 1.
  3. 3 cos 60° = 3 × (1/2) = 3/2.
  4. Combine: 1 + 3/2 − 1 = 3/2.
  5. Check each substitution against the table once more: sin 30° = 1/2 and cos 60° = 1/2 are the middle mirror pair, tan 45° = 1 is the centre of the tangent row. The answer 3/2 stands.

Common Misconceptions

  • tan 90° is not infinity. It is undefined, because cos 90° = 0 and you cannot divide by zero.
  • sin 30° = 1/2, not √3/2. The √3/2 belongs to sin 60° and cos 30°; mixing the pair is the classic slip.
  • Degrees and radians are different scales. sin(π/6) = 1/2 matches sin 30° only because π/6 radians equals 30°.

Real-World Applications

  • Geometry problems: exact side lengths in triangles with standard angles, no calculator needed.
  • Physics: resolving forces at 30°, 45° and 60°, the most common inclined-plane angles.
  • Engineering: quick mental checks of designs involving standard slopes.
  • Calculus: evaluating limits and derivatives at standard points exactly.

The standard angles 0°, 30°, 45°, 60° and 90° carry exact values built on the √n/2 pattern for sine, mirrored for cosine. Memorize the table, derive it from the two special triangles when in doubt, and never divide by cos 90°.

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