Solving for Missing Sides and Angles

Unit 8: Right Triangles and Trigonometry · ~15–30 min

Solving for a missing side

To find a missing side, pick the ratio (sin, cos, or tan) that connects the angle you know to the side you know and the side you want. Set up the equation, then solve for the unknown — usually by cross-multiplying or dividing.

• If the unknown is in the numerator's position (e.g. opposite in sin), multiply both sides by the known side.
• If the unknown is in the denominator's position (e.g. hypotenuse in sin), solve using division after cross-multiplying.

Solving for a missing angle

If you know two sides but not the angle, use the inverse trig functions — \(sin^{-1}\), \(cos^{-1}\), \(tan^{-1}\) (labeled \(sin^{-1}\), etc. on a calculator) — to undo the ratio and recover the angle itself. For example, if tan θ = 0.75, then θ = \(tan^{-1}\)(0.75).

Guided practice

• 1.A right triangle has a 52° angle and hypotenuse 14. Find the side opposite the 52° angle.
  Show answer

  sin 52° = opp/14, so opp = 14 · sin 52° ≈ 14(0.7880) ≈ 11.0.
• 2.A right triangle has a 20° angle and the adjacent side is 15. Find the hypotenuse.
  Show answer

  cos 20° = 15/hyp, so hyp = 15/cos 20° ≈ 15/0.9397 ≈ 16.0.
• 3.A right triangle has legs 7 (opposite θ) and 24 (adjacent). Find θ to the nearest degree.
  Show answer

  tan θ = 7/24 ≈ 0.2917, so θ = \(tan^{-1}\)(0.2917) ≈ 16°.
• 4.A right triangle has hypotenuse 30 and one leg 18 (opposite θ). Find θ to the nearest degree.
  Show answer

  sin θ = 18/30 = 0.6, so θ = \(sin^{-1}\)(0.6) ≈ 37°.
• 5.A right triangle has hypotenuse 12 and a 40° angle. Find both legs, rounded to one decimal.
  Show answer

  Opposite = 12 · sin 40° ≈ 12(0.6428) ≈ 7.7. Adjacent = 12 · cos 40° ≈ 12(0.7660) ≈ 9.2.