Solving Right Triangles

One side and one angle

To solve a triangle is to find all of its sides and angles. In a right triangle, one acute angle and one side are enough. The angle fixes the shape, because every right triangle with that angle is similar to every other. The side fixes the size.

Here the hypotenuse is 10 and one angle is 30°. The other acute angle is 90 − 30 = 60°. The side across from the 30° angle, the opposite side, is missing.

A right triangle with a hypotenuse of 10 and an angle of 30°. The side across from the 30° angle is unknown.

Choose the ratio

Name the sides from the angle you are given. Then pick the ratio that uses exactly two sides: the side you know and the side you want. Opposite and hypotenuse make the sine, adjacent and hypotenuse make the cosine, and opposite and adjacent make the tangent. The third side is not in it.

Here the known side is the hypotenuse and the wanted side is the opposite, so the ratio is the sine: sin 30° = opposite/10. Multiply both sides by 10 to get the unknown on its own: opposite = 10 sin 30°.

$\sin 30^{\circ }=\frac{1}{2}$sin 30° = 1/2 exactly, so the opposite side is $10\times \frac{1}{2}=5$10 × 1/2 = 5.

The opposite side is 10 sin 30° = 5, half of the hypotenuse.

The unknown underneath

Sometimes the side you want is in the denominator of the ratio. A right triangle has an angle of 60°, the side adjacent to it is 12, and the hypotenuse is wanted. Adjacent and hypotenuse make the cosine: cos 60° = 12/hypotenuse.

Multiply both sides by the hypotenuse: hypotenuse × cos 60° = 12. Then divide both sides by cos 60°: hypotenuse $=12/\cos 60^{\circ }$= 12 / cos 60°. $\cos 60^{\circ }=\frac{1}{2}$cos 60° = 1/2, and $12÷\frac{1}{2}=24$12 ÷ 1/2 = 24, so the hypotenuse is 24.

So when the unknown is on top, multiply the known side by the ratio; when it is underneath, divide the known side by the ratio. Check the result for size: the hypotenuse must be the longest side, and 24 is longer than 12.

An angle of 60° with an adjacent side of 12. The hypotenuse is $12/\cos 60^{\circ }=24$12 / cos 60° = 24.

A full solution

A right triangle has an angle of 40°, and the side adjacent to it is 15 m. Find the other two sides and the other angle, giving lengths to two decimal places.

The other angle is 90 − 40 = 50°.

For the opposite side, the known side is adjacent and the wanted side is opposite, so use the tangent: tan 40° = opposite/15. The unknown is on top, so multiply: opposite = 15 tan 40° = 15 × 0.83910… = 12.586… m, which is 12.59 m to two decimal places.

For the hypotenuse, use the side that was given, not the one just worked out, so that a slip in one answer cannot spread to the next. Adjacent and hypotenuse make the cosine: cos 40° = 15/hypotenuse. The unknown is underneath, so divide: hypotenuse $=15/\cos 40^{\circ }$= 15 / cos 40°. That is 15 ÷ 0.76604… = 19.581…, so the hypotenuse is 19.58 m to two decimal places.

Keep the full calculator values while working, and round only the final answers. Rounding early, such as using 0.84 for tan 40°, gives 15 × 0.84 = 12.60 m, which is wrong in the last figure.

The solved triangle: an angle of 40° with an adjacent side of 15 m, an opposite side of 12.59 m and a hypotenuse of 19.58 m, to two decimal places.

Check with Pythagoras

Pythagoras’ theorem checks the sides without using the angle at all. With the unrounded values, $12.586^2+15^2=158.42+225=383.42$12.586² + 15² = 158.42 + 225 = 383.42, and $19.581^2=383.42$19.581² = 383.42. The two agree, so the sides fit together.

For the first triangle, the third side is $10\cos 30^{\circ }=10\times \frac{\sqrt{3}}{2}=5\sqrt{3}$10 cos 30° = 10 × √3/2 = 5√3. Then $5^2+{\left(5\sqrt{3}\right)}^2=25+75=100=10^2$5² + (5√3)² = 25 + 75 = 100 = 10², as it should be.

The usual mistakes

Dividing when the unknown is on top. $10/\sin 30^{\circ }=20$10 / sin 30° = 20 would make the opposite side longer than the hypotenuse, which is impossible. Since sin 30° = opposite/10, the opposite side is 10 × sin 30°.

Using the ratio for the wrong side. With the hypotenuse known at 30°, the adjacent side needs the cosine: $10\cos 30^{\circ }=5\sqrt{3}$10 cos 30° = 5√3. The 5 that the sine gives is the opposite side.

Reading sin 30° as 1. Only the sine of 90° is 1; sin 30° is $\frac{1}{2}$1/2.

Rounding before the last step. Keep the calculator’s full values and round the answer once.

An angle of depression

In the application below, the lighthouse keeper looks down at a boat. The angle between the horizontal line through the keeper’s eye and the line of sight down to the boat is called the angle of depression.

The horizontal through the lamp and the sea are parallel, and the line of sight crosses both, so the angle of depression at the lamp and the angle at the boat, between the sea and the line of sight, are alternate angles. They are equal, so the angle of depression is also an angle inside the right triangle, at the boat.