Trig Ratios (SOH–CAH–TOA)

TipLearning Objectives
  • Define the sine, cosine, and tangent ratios in right triangles.
  • Identify opposite, adjacent, and hypotenuse relative to an acute angle.
  • Evaluate trig ratios for common right triangles.
  • Apply SOH–CAH–TOA to solve for missing sides or angles.

Key Ideas

In a right triangle, for an acute angle \(\theta\):

  • Sine: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
  • Cosine: \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
  • Tangent: \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]

“SOH–CAH–TOA” is the memory aid.

Right triangle labeled with opposite, adjacent, and hypotenuse.

Common Problem Types

Identifying Opp/Adj/Hyp

Given a marked angle, identify each side relative to that angle.

Evaluating Trig Ratios

Example: \(\sin\theta = 3/5\) in a triangle with opposite 3, hypotenuse 5.

Solving for a Missing Side

Use \(\sin\), \(\cos\), or \(\tan\) and multiply/divide accordingly.

Finding an Angle from a Ratio

Use inverse trig (e.g., \(\theta = \sin^{-1}(0.6)\)) when needed.

Using Special Triangles

Use \(30^\circ\)–\(60^\circ\)–\(90^\circ\) and \(45^\circ\)–\(45^\circ\)–\(90^\circ\) ratios when applicable.

Strategies

  • Always anchor the sides relative to the reference angle.
  • Use the ratio that involves the given side and the unknown side.
  • Draw a quick sketch if the triangle isn’t shown.
  • Leave answers in exact form unless decimals are required.

Worked Examples

Example 1

In a right triangle, \(\theta\) has opposite side 8 and hypotenuse 17.
\[ \sin\theta = \frac{8}{17} \]

Example 2

Find \(x\) if \(\cos 30^\circ = \frac{x}{10}\). \[ x = 10\cos 30^\circ = 10\left(\frac{\sqrt3}{2}\right)=5\sqrt3 \]

WarningCommon Mistakes
  • Mixing up opposite and adjacent.
  • Using the wrong reference angle.
  • Forgetting that trig ratios apply only in right triangles (for now).
  • Rounding too early.

Practice Problems

  1. If \(\sin\theta = 4/5\), what is the opposite/hypotenuse ratio?
  2. In a right triangle, \(\tan\theta = 3/4\) and adjacent = 12. Find opposite.
  3. Solve: \(\cos\theta = 0.6\).
  4. A ladder leans against a wall with angle \(60^\circ\) to the ground. Height if ladder is 12 ft?
  1. Opp = 4, Hyp = 5
  2. \(\tan\theta = \frac{\text{opp}}{12}=3/4\) → opp = 9
  3. \(\theta = \cos^{-1}(0.6)\)
  4. \(12\sin60^\circ = 12(\sqrt3/2)=6\sqrt3\)

Summary

  • Use SOH–CAH–TOA to find missing sides or angles.
  • Opp/Adj depend on the chosen angle.
  • Special triangles help produce exact values.
  • Identify the reference angle first.
  • Use the ratio that uses the known and unknown sides.
  • Leave radicals in simplified exact form when possible.