LEMMA

Trigonometry

Right-triangle trigonometry

ALEKS placement

SOH-CAH-TOA relates one acute angle to two sides.

What this covers

sinθ=hypopp​,cosθ=hypadj​,tanθ=adjopp​,a2+b2=c2
  • If a side is missing, start with the Pythagorean theorem: the squares of the two shorter sides add to the square of the longest side.
  • For shorter sides 3 and 4, that gives 9 + 16 = 25, so the longest side is 5.
  • Label the sides relative to the angle you are given, not to the page.
  • Pick the ratio that uses the side you know and the side you want.
  • Solve the resulting proportion.
  • A 45-45-90 triangle has side ratio 1 : 1 : √2.
  • A 30-60-90 triangle has side ratio 1 : √3 : 2, opposite 30°, 60°, and 90°.
  • Use an inverse trig function when the angle itself is the unknown.

Worked example

Worked example

A ramp rises 3 ft over a 12 ft horizontal run. Find its angle.

  1. tanθ=123​Opposite over adjacent, since neither side is the hypotenuse.
  2. θ=arctan(0.25)≈14∘Apply the inverse tangent.
Another worked example

The side opposite a 30° angle is 4. Find the hypotenuse h.

  1. sin30∘=h4​Sine connects the opposite side and the hypotenuse.
  2. 21​=h4​⇒h=8The opposite side is half the hypotenuse in a 30-60-90 triangle.

A common mistake

A mistake Lemma catches

From sin60∘=10o​=2√3​

o=√3/210​o=102√3​

“sin 60° = opposite/hypotenuse = o/10, so multiply both sides by 10: o = 10 · √3/2 = 5√3 ≈ 8.66. Dividing 10 by √3/2 gives 20/√3 ≈ 11.5 — longer than the hypotenuse 10, which no leg can be.”

Try one

Sample problem

A ladder, wall, and ground form the shown right triangle. For ground angle θ, the wall height is 9 and the ladder is 41. Find sin θ.

sin(θ)=?

Practice right triangles free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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