DWG NO. 4.3 — Lesson 3 of 6

Right-Triangle Trigonometry Revisited

Unit 4: Trigonometric Functions and the Unit Circle · ~20 min

Right-triangle trigonometry probably isn't new — but it's worth revisiting now that θ is defined by the unit circle. For an acute angle θ in a right triangle, drop the triangle so its vertex sits at the origin and the adjacent leg lies along the positive x-axis: the hypotenuse becomes a radius, and if that radius is scaled to length 1, the triangle's own legs become exactly cos θ and sin θ. The familiar ratios and the unit-circle coordinates are the same idea, just scaled differently.

• sin θ = \(\frac{opposite}{hypotenuse}\)
• cos θ = \(\frac{adjacent}{hypotenuse}\)
• tan θ = \(\frac{opposite}{adjacent}\)  (also tan θ = sin \(\frac{θ}{cos}\) θ)
• csc θ = \(\frac{1}{sin}\) θ, sec θ = \(\frac{1}{cos}\) θ, cot θ = \(\frac{1}{tan}\) θ — the three reciprocal functions

Solving right triangles

"Solving" a right triangle means finding every missing side and angle. If you know one side and one acute angle, or any two sides, the three ratios (and the Pythagorean theorem, when needed) are enough to find everything else. A common application is an angle of elevation or angle of depression — the angle between a horizontal line of sight and a line up to (or down to) an object.

Guided practice

• 1.A right triangle has hypotenuse 20 and an acute angle of 40°. Find the side adjacent to that angle.
  Show answer

  adjacent = 20 · cos 40° ≈ 20(0.766) ≈ 15.3.
• 2.A right triangle has opposite leg 7 and hypotenuse 25. Find θ.
  Show answer

  sin θ = \(\frac{7}{25}\) = 0.28, so θ = sin⁻¹(0.28) ≈ 16.3°.
• 3.Write cot θ, sec θ, and csc θ as ratios of opposite, adjacent, and hypotenuse.
  Show answer

  cot θ = \(\frac{adjacent}{opposite,}\) sec θ = \(\frac{hypotenuse}{adjacent,}\) csc θ = \(\frac{hypotenuse}{opposite}\) — each is the reciprocal of tan, cos, and sin respectively.
• 4.From a point 60 ft from a building's base, the angle of elevation to the roof is 38°. Find the building's height.
  Show answer

  height = 60 · tan 38° ≈ 60(0.781) ≈ 46.9 ft.
• 5.Explain why sin θ = cos(90° − θ) for any acute angle θ, using the two non-right angles of a right triangle.
  Show answer

  The two acute angles of a right triangle sum to 90°, so if one is θ the other is 90° − θ. The side opposite θ is adjacent to the other angle, so sin θ (\(\frac{opposite}{hypotenuse}\) for θ) equals cos(90° − θ) (\(\frac{adjacent}{hypotenuse}\) for the complementary angle) — they describe the same side ratio from two different corners.