SOH CAH TOA Calculate an angle

We can use the formulas SOH CAH TOA to calculate an angle when we know:

  • the triangle is right angled
  • at least two of the three sides

Here are the SOH CAH TOA formulas we use to calculate a side:

    \[\sin (\theta)=\frac{\text{opp}}{\text{hyp}}; \quad \cos (\theta)=\frac{\text{adj}}{\text{hyp}}; \quad \tan (\theta)=\frac{\text{opp}}{\text{adj}}\]

We use the inverse trig functions to calculate an angle:

    \[\theta=\sin^{-1}\Big(\frac{\text{opp}}{\text{hyp}}\Big); \quad \theta=\cos^{-1}\Big(\frac{\text{adj}}{\text{hyp}}\Big); \quad \theta=\tan^{-1}\Big(\frac{\text{opp}}{\text{adj}}\Big)\]

The inverse trig functions can be found above the regular trig functions. Press ‘shift’ or ‘2nd’ on your calculator to access these functions.

inverse trig functions on a scientific calculator

Example

Right angle triangle showing two sides

The blue side 16.6 is opposite the angle \theta. The red side 19.8 is the hypotenuse.

Knowing ‘opp’ and ‘hyp’ we can use

\theta=\sin^{-1}\Big(\frac{\text{opp}}{\text{hyp}}\Big)

Right angled triangle showing two sides and calculation for angle

Practice using SOH CAH TOA to calculate an angle

  1. estimate the angle.
  2. draw the triangle,
  3. label the sides ‘opp’, ‘adj’ and ‘hyp’
  4. select the correct trig ratio to solve the problem
  5. make sure your calculation matches your initial estimation.

GeoGebra link


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