Trigonometric Ratios: SOHCAHTOA

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Skill: trig-ratios

Trigonometric Ratios: SOHCAHTOA

Three ratios connect an acute angle to the sides of a right triangle. Learn to label first, pick the right ratio, and turn angles into distances.

1 The Three Ratios

Pick an acute angle θ. The hypotenuse is always opposite the right angle. The other two sides are named relative to θ: opposite is across from θ, adjacent touches θ.

SINE
sin θ = opp / hyp
Sine = Opposite / Hypotenuse
COSINE
cos θ = adj / hyp
Cosine = Adjacent / Hypotenuse
TANGENT
tan θ = opp / adj
Tangent = Opposite / Adjacent

Two directions, one tool. Given an angle, find a side (plug into the ratio and solve). Given two sides, find an angle — use the inverse trig button: θ = sin¹(ratio). A ratio is not an angle; the inverse undoes the ratio.

Opposite and adjacent swap when you switch which acute angle you work with — but the hypotenuse never changes. Always label the sides before you choose a ratio.

2 See It

Label first, then choose your ratio — and angle-of-elevation stories are right triangles standing up.

adjacent opposite hypotenuse θ SOH · CAH · TOA
Opposite and adjacent swap depending on which acute angle you pick — but the hypotenuse never changes. Label first, then choose your ratio.
35° observer tree sight line
Angle of elevation problems are right triangles standing up. The trig ratio you need depends on which two sides the story gives you.

3 Special Angles Worth Memorizing

At 30°, 45°, and 60° the trig values are exact — no calculator needed. These power the “exact form” answers.

θsin θcos θtan θ
30°1/2√3/2√3/3
45°√2/2√2/21
60°√3/21/2√3

Read them off the two special triangles: the 30-60-90 has sides 1, √3, 2 and the 45-45-90 has sides 1, 1, √2. Notice sin 30° = cos 60° — cofunctions of complementary angles always match.

4 Worked Examples

Follow each step. The pattern never changes: label the sides, pick the ratio, solve.

Example 1 Find the ratio: opposite 3, hypotenuse 5
  1. Label: the side across from θ is the opposite (3); the slanted side is the hypotenuse (5).
  2. Opposite and hypotenuse means sine: sin θ = 3/5.
  3. Divide: sin θ = 0.6.
sin θ = 0.6. Check: a 3-4-5 triangle, and sine is always below 1. ✓
Example 2 Find a side: tan 30° = x/12
  1. Multiply both sides by 12: x = 12 · tan 30°.
  2. Exact value: tan 30° = √3/3, so x = 12 · √3/3 = 4√3.
  3. Decimal: x ≈ 6.93.
x = 4√3 ≈ 6.93. Check: tan 30° ≈ 0.577, and 12 × 0.577 ≈ 6.93. ✓
Example 3 Find an angle: cos θ = 8/17
  1. A ratio is not an angle — undo cosine with inverse cosine: θ = cos¹(8/17).
  2. Calculator in degree mode: θ ≈ 61.93°.
θ ≈ 61.93°. Check: cos 61.93° ≈ 0.4706 = 8/17. ✓
Example 4 Word problem: a 10-ft ladder makes 70° with the ground
  1. Draw it: the height is opposite the 70° angle; the ladder is the hypotenuse.
  2. Opposite and hypotenuse means sine: sin 70° = h/10.
  3. Solve: h = 10 · sin 70° ≈ 10 × 0.9397 ≈ 9.40 ft.
h ≈ 9.40 ft. Check: less than the 10-ft ladder, and a steep 70° should reach nearly the full length. ✓

5 Common Mistakes

Three traps that catch nearly everyone. Learn to spot them here instead of on a test.

Mistake 1: Mixing up opposite and adjacent
Wrong
Using the side next to θ as “opposite” in sine — because it looks across from where you’re sitting.
Right
Opposite is across from θ; adjacent touches θ (but isn’t the hypotenuse). Physically point at the angle, then point across the triangle — that’s opposite.
Fix: label the three sides before you touch the calculator. Labels first, ratio second.
Mistake 2: Calculator in radian mode
Wrong
Typing sin(30) and getting −0.988 — then building a whole word problem on it.
Right
sin 30° = 0.5. Look for the “D” (degree) indicator before you compute.
Fix: if a trig answer looks absurd, check degree mode before anything else. This mistake has sunk thousands of tests.
Mistake 3: sin θ = 0.6 → θ = 0.6
Wrong
Dropping the inverse: treating the ratio 0.6 as if it were already the angle.
Right
θ = sin¹(0.6) ≈ 36.87°. A ratio is not an angle.
Fix: to go from a ratio back to an angle you need the inverse button (sin¹, cos¹, tan¹).

6 Quick Checks

Cover the answer, try it, then reveal.

1. sin θ with opposite 5 and hypotenuse 13.
Answer
sin θ = 5/13 ≈ 0.385.
2. Find θ (acute): tan θ = 1.
Answer
θ = 45° (the 45-45-90 triangle has opposite = adjacent).
3. Find the hypotenuse h: cos 60° = 4/h.
Answer
cos 60° = 1/2, so 1/2 = 4/h and h = 8.

7 Key Points

  • sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj — SOHCAHTOA.
  • Opposite and adjacent are relative to your angle; the hypotenuse never changes.
  • Angle → side: plug into the ratio and solve. Sides → angle: use the inverse (θ = sin¹(ratio)).
  • Memorize 30°/45°/60°: sin 30° = 1/2, tan 30° = √3/3, sin 45° = √2/2, tan 45° = 1, cos 60° = 1/2.
  • Calculator in degree mode — check for the “D” before every trig computation.
  • Label the sides first. Most trig errors are labeling errors, not calculator errors.
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