SOH CAH TOA & Finding a Side

Section 1 of 2

Labelling a Right-Angled Triangle

The three sides

The hypotenuse is the longest side, opposite the right angle. The opposite faces the angle you are using. The adjacent is beside it.

SOH CAH TOA

$\sin A = \dfrac{\text{opp}}{\text{hyp}}$, $\cos A = \dfrac{\text{adj}}{\text{hyp}}$, $\tan A = \dfrac{\text{opp}}{\text{adj}}$. Remember: Silly Old Harry Caught A Herring Trawling Off America.

Section 2 of 2

Finding a Side

Which ratio?

Mark the angle, then label the two sides that matter (the one you have and the one you want). Pick the ratio that uses both.

Worked example — hypotenuse and the angle

A right-angled triangle has hypotenuse $12$ and an angle of $30^\circ$. Find the opposite side $x$.

Have hyp, want opp $\Rightarrow$ use $\sin$

$\sin 30 = \dfrac{x}{12} \Rightarrow x = 12\sin 30$

$x = 6$

Worked example — finding the adjacent

The hypotenuse is $20$ and the angle is $47^\circ$. Find the adjacent side $x$.

Have hyp, want adj $\Rightarrow$ use $\cos$

$\cos 47 = \dfrac{x}{20} \Rightarrow x = 20\cos 47$

$x = 13.63 = 13.6$

$x = 13.6$

Worked example — using tan

The adjacent side is $12$ and the angle is $41^\circ$. Find the opposite side $x$.

Have adj, want opp $\Rightarrow$ use $\tan$

$\tan 41 = \dfrac{x}{12} \Rightarrow x = 12\tan 41$

$x = 10.43 = 10.4$

$x = 10.4$

You try

Hypotenuse $15$, angle $25^\circ$. Find the adjacent side $x$.

Have hyp, want adj ⇒ $\cos$.

$x = 15\cos 25 = 13.59 = 13.6$

$x = 13.6$

You try

Adjacent $16$, angle $31^\circ$. Find the opposite side $x$.

Have adj, want opp ⇒ $\tan$.

$x = 16\tan 31 = 9.61 = 9.6$

$x = 9.6$

You try

Hypotenuse $12$, angle $43^\circ$. Find the opposite side $x$.

Have hyp, want opp ⇒ $\sin$.

$x = 12\sin 43 = 8.18 = 8.2$

$x = 8.2$

You try

Hypotenuse $19$, angle $21^\circ$. Find the adjacent side $x$.

$\cos$.

$x = 19\cos 21 = 17.7$

$x = 17.7$

You try

Hypotenuse $9$, angle $27^\circ$. Find the opposite side $x$.

$\sin$.

$x = 9\sin 27 = 4.08 = 4.1$

$x = 4.1$

You try

Adjacent $8$, angle $39^\circ$. Find the opposite side $x$.

$\tan$.

$x = 8\tan 39 = 6.47 = 6.5$

$x = 6.5$

That’s Class 1.

Labelling, SOH CAH TOA, and finding a side when $x$ is on top. Class 2: when $x$ is underneath.