SOH CAH TOA Calculate a side

To use SOH CAH TOA to calculate a side on a right angled triangle, we first have to label the sides.

label the hypotenuse

The hypotenuse always lies opposite the right angle.

The two shorter sides are the ‘opp’ and ‘adj’. The names ‘opp’ and ‘adj’ refer to one of the angles.

label the opposite side

label the adjacent side

The other acute angle isn’t labeled in these examples. The blue side is adjacent to the other angle, and the green side is opposite. This video explains well.

Use SOH CAH TOA to calculate a side on a right angled triangle

We can calculate a side on a triangle with SOH CAH TOA when we know

  • the triangle is right angled
  • the size of at least one acute angle
  • the length of at least one side

SOH CAH TOA refers to these formulas:

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

A simple SOH CAH TOA problem presents exactly the information you need and no more.

GeoGebra link

Practice

Use the sine; cosine or tangent ratio to find the side indicated.

Draw the triangle onto paper. Label, and show working clearly.

GeoGebra link

 

Why do the three formula work?

Review Introduction to Trigonometry to see where sine, cosine and tangent values come from.

The sine of an angle can be defined as the side opposite the angle on a right angle triangle that has hypotenuse length 1. The cosine is the length of the adjacent side, and the tangent is the length on the tangent as seen in the diagram:

right triangle with hypotenuse 1

For every other similar right angled triangle, we see that the scale factor of enlargement is nothing other than the length of the hypotenuse:

right triangle with hypotenuse r

In this diagram, r = \text{ hyp }.

Focus on the side opposite the marked angle, the green side. We see that

\text{opposite }=\text{ hypotenuse }\times \sin(24°)

or, in general,

\text{opp }=\text{ hyp }\times \sin(\theta)

Which leads us to the formula:

\frac{\text{opp }}{\text{ hyp }}=\sin(\theta)

Similarly, the adjacent side, the red length can be calculated as:

\text{adjacent }=\text{ hypotenuse }\times \cos(24°)

Which leads to the formula

\frac{\text{adj }}{\text{ hyp }}=\cos(\theta)

Finally, through examining the triangle made with the tangent line:

tangent dimension on a circle radius r

We see that in this case, the side opposite the angle divided by the side adjacent gives us \tan(\theta):

\frac{\text{opp}}{\text{adj}}=\frac{r \tan (\theta)}{r}=\tan (\theta)

This leads us to three tidy trig formulae for right angled triangles:

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

Which leads us to the mnemonic: SOH CAH TOA, used for calculating sides and angles on any right angled triangle.


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