The three ratios

Pick an acute angle \(\theta\) in a right triangle. Name the sides relative to \(\theta\):

  • Hypotenuse (H): opposite the right angle, always the longest side.
  • Opposite (O): across from \(\theta\).
  • Adjacent (A): next to \(\theta\), not the hypotenuse.

Then:

$$ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} $$

Mnemonic: SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent).

Tangent is also the ratio of sine to cosine:

$$ \tan\theta = \frac{\sin\theta}{\cos\theta} $$

Worked example 1: find a side

A ladder leans at 65° to the ground; its foot is 2 m from the wall. How long is the ladder?

The known side (2 m) is adjacent to 65°, the unknown (ladder) is the hypotenuse → use cosine:

$$ \cos 65^\circ = \frac{2}{H} \quad\Rightarrow\quad H = \frac{2}{\cos 65^\circ} \approx \frac{2}{0.4226} \approx 4.73 \text{ m} $$ 
import math
H = 2 / math.cos(math.radians(65))
print(round(H, 2))  # 4.73

Worked example 2: find an angle

A 5 m ladder reaches 4 m up a wall. What angle does it make with the ground?

Opposite = 4, hypotenuse = 5 → use sine, then invert:

$$ \sin\theta = \frac{4}{5} = 0.8 \quad\Rightarrow\quad \theta = \arcsin(0.8) \approx 53.13^\circ $$ 
import math
theta = math.degrees(math.asin(4 / 5))
print(round(theta, 2))  # 53.13
const theta = Math.asin(4 / 5) * 180 / Math.PI;
console.log(theta.toFixed(2)); // 53.13

Reciprocal functions

Three more functions are just reciprocals (less common, but you will see them in docs):

$$ \csc\theta = \frac{1}{\sin\theta}, \qquad \sec\theta = \frac{1}{\cos\theta}, \qquad \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} $$

Solving checklist

  1. Draw the triangle, mark the right angle.
  2. Label O / A / H relative to your angle.
  3. Choose the ratio containing the known + unknown (SOH-CAH-TOA).
  4. Solve algebraically; convert degrees ↔ radians before calling sin/cos/tan.
  5. Sanity-check: hypotenuse must be longest; angles sum to 180°.
import math

def solve_missing_side(known, angle_deg, ratio="opp-hyp"):
    """Demo solver: given one side + acute angle, return another side."""
    r = math.radians(angle_deg)
    if ratio == "opp-hyp":      # sin = opp/hyp -> hyp = opp/sin
        return known / math.sin(r)
    if ratio == "adj-hyp":      # cos = adj/hyp -> hyp = adj/cos
        return known / math.cos(r)
    if ratio == "opp-adj":      # tan = opp/adj -> opp = adj*tan
        return known * math.tan(r)
    raise ValueError("unknown ratio")

print(round(solve_missing_side(2, 65, "adj-hyp"), 2))  # 4.73 (ladder example)

Practice

  1. Right triangle: angle 30°, hypotenuse 10. Find opposite and adjacent. (Answers: 5 and ≈ 8.66.)
  2. Opposite = 7, adjacent = 24. Find the hypotenuse and both acute angles. (Hint: Pythagoras first: \(h = 25\).)
  3. A ramp rises 1 m over 12 m horizontal. What is its angle? Why does atan(1/12) give it directly?
  4. Code it: write height_from_shadow(shadow_len, sun_elevation_deg) using tangent.

Next: The Unit Circle — extending these ratios past 90°.