Trigonometry to Find Lengths Flashcards

All 6 cards in this deck

What are the three trigonometric ratios for a right-angled triangle?

$\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$ (SOH CAH TOA).

How are the three sides of a right-angled triangle labelled relative to a marked angle $\theta$?

Hypotenuse: opposite the right angle (longest side). Opposite: across from $\theta$. Adjacent: between $\theta$ and the right angle.

Which trigonometric ratio do you use when you know the hypotenuse and want the side opposite the given angle?

e.g. hypotenuse $178$ mm, angle $34^\circ$, opposite side $x$

Sine: $\sin 34^\circ = \frac{x}{178}$, so $x = 178 \times \sin 34^\circ$.

What are the steps to find a missing side using trigonometry?

e.g. angle $56^\circ$, adjacent side $12$ cm, find the opposite side $BC$

1. Label the sides for $56^\circ$: $12$ is adjacent, $BC$ is opposite
2. Choose the ratio: $\tan 56^\circ = \frac{BC}{12}$
3. Rearrange and work out: $BC = 12 \times \tan 56^\circ = 17.8$ cm (1 d.p.)

What do you do when the unknown side is the denominator of the trig ratio?

e.g. $\sin 30^\circ = \frac{5}{x}$

Divide the known side by the ratio: $x = 5 \div \sin 30^\circ = 10$.

What are the steps to find the area of a right-angled triangle when you know the base and one acute angle?

e.g. base $10$ cm, angle $45^\circ$ between base and hypotenuse

1. Set up the ratio for the height: $\tan 45^\circ = \frac{h}{10}$
2. Find the height: $h = 10 \times \tan 45^\circ = 10$ cm
3. Area $= \frac{1}{2} \times 10 \times 10 = 50$ cm$^2$