Apply the Pythagorean theorem, the special right triangles, and the sine, cosine and tangent ratios (SOH-CAH-TOA) to find sides and angles of right triangles (Geometry).

What this topic is asking

Right-triangle trigonometry is a high-frequency ACT topic. It tests the Pythagorean theorem, the two special right triangles, common Pythagorean triples, and the three trig ratios sine, cosine and tangent (SOH-CAH-TOA) for finding missing sides and angles.

The Pythagorean theorem

The cornerstone of right-triangle geometry.

Memorising the common triples saves time: $3$-$4$-$5$, $5$-$12$-$13$, $8$-$15$-$17$, $7$-$24$-$25$, and any multiple (such as $6$-$8$-$10$). Spotting a triple lets you read the missing side without computing a square root.

The special right triangles

Two triangles have fixed side ratios worth memorising.

So a 45-45-90 triangle with legs 5 has hypotenuse $5\sqrt{2}$, and a 30-60-90 triangle with short leg 4 has long leg $4\sqrt{3}$ and hypotenuse 8.

The trig ratios: SOH-CAH-TOA

The three ratios relate an acute angle to the sides.

The mnemonic SOH-CAH-TOA encodes the three ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Choosing the right tool

A quick decision guides every right-triangle problem: if you have two sides and want the third, use the Pythagorean theorem; if an angle is involved (given or wanted) along with sides, use a trig ratio. To find an unknown angle from two sides, use an inverse trig function (for example, $\theta = \tan^{-1}\frac{\text{opp}}{\text{adj}}$ on a calculator). Recognising a special triangle or a triple short-circuits the computation entirely.

Angles of elevation and depression

A common application dresses a right triangle as a real scene: the angle of elevation is measured upward from the horizontal to a line of sight, and the angle of depression downward. A problem like "from 50 feet away, the angle of elevation to the top of a tree is $40°$; how tall is the tree?" sets up $\tan 40° = \frac{\text{height}}{50}$, so height $= 50\tan 40°$. Drawing the right triangle with the horizontal distance, the vertical height, and the line of sight as the hypotenuse turns the words into a single trig equation.

Try this

Q1. A right triangle has legs 9 and 12. Find the hypotenuse. [1 point]

• Cue. $9^{2} + 12^{2} = 81 + 144 = 225$, so $c = 15$ (a 3-4-5 triple times 3).

Q2. In a 45-45-90 triangle, each leg is 7. Find the hypotenuse. [1 point]

• Cue. Hypotenuse is leg times $\sqrt{2}$: $7\sqrt{2}$.