Trigonometric Ratios of Special Angles

• Trigonometric ratios


  Ratios that relate the sides of a right triangle to the angles of the triangle
• Finding trigonometric ratios of special angles

  1. Identify the special angle (30°, 45°, 60°)

  2. Draw the right triangle with the special angle

  3. Apply the formulas for sine, cosine, tangent, cosecant, secant, cotangent

  4. Simplify the ratios
• Equilateral triangle

  • All sides are equal
  • All angles are 60°
• Perpendicular bisector


  Line that is perpendicular to and bisects a line segment
• In an equilateral triangle, the perpendicular bisector divides the side into two equal parts
• Finding trigonometric ratios for 30° and 60° angles

  1. Identify the sides of the right triangle (a, a/2, √3a/2)

  2. Apply the formulas: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, csc 30° = 2, sec 30° = 2√3/3, cot 30° = √3

  3. Apply the formulas: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3, csc 60° = 2/√3, sec 60° = 2, cot 60° = 1/√3
• Finding trigonometric ratios for 45° angles


  1. Identify the sides of the right triangle (a, a, a√2)

  2. Apply the formulas: sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1, csc 45° = √2, sec 45° = √2, cot 45° = 1
• Complementary angles (45°, 45°, 90°) have equal trigonometric ratios
• Hypotenuse


  The longest side of a right-angled triangle, opposite the right angle
• Legs


  The two shorter sides of a right-angled triangle
• Finding trigonometric ratios

  1. Select the appropriate ratio

  2. Substitute the given values

  3. Simplify the expression
• Trigonometric ratios of special angles (0°, 30°, 45°, 60°, 90°) can be memorised
• Solving for unknown angles

  1. Use the given trigonometric ratio

  2. Substitute the known values

  3. Solve for the unknown angle
• In a right-angled triangle, the sum of the angles is 180°
• Trigonometric ratios are related: sin²θ + cos²θ = 1, tan θ = sin θ / cos θ