learn › Trigonometry

Right-triangle trig

lesson · about 3 minutes

Sine, cosine, and tangent as side ratios.

Pen and paper is fine · no calculator needed why?

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the idea

In a right triangle, call one of the two smaller angles A. The hypotenuse is the longest side, across from the right angle. The other two sides are the leg opposite A and the leg next to A.

Sine, cosine and tangent are ratios of these sides. They depend only on the angle, not on the size of the triangle, so one table of values works for every triangle.

techniques

SOH CAH TOA

You know the sides and want sin A, cos A or tan A.

SOH: sine is opposite ÷ hypotenuse.
CAH: cosine is the leg next to A ÷ hypotenuse.
TOA: tangent is opposite ÷ the leg next to A.
Reduce the fraction to lowest terms.

worked example

Example: In a right triangle, the leg opposite angle A is 10, the leg next to angle A is 24, and the hypotenuse is 26. What is cos A? Give a fraction.

Cosine is the leg next to A ÷ hypotenuse.
24/26 = 12/13.

Answer: 12/13

Multiply the hypotenuse

You know the hypotenuse and a 30°, 45° or 60° angle.

Opposite leg: hypotenuse × sine of the angle. Next-to leg: hypotenuse × cosine.
sin 30° and cos 60° are 1/2. sin 45° and cos 45° are √2/2, about 0.7071.
sin 60° and cos 30° are √3/2, about 0.8660. Round only at the end.

worked example

Example: A right triangle has hypotenuse 10 and a 30° angle. How long is the leg next to the 30° angle? Round to 2 decimal places.

The leg next to the angle uses cosine.
cos 30° = √3/2, about 0.8660.
10 × 0.8660 = 8.66.

Answer: 8.66

Read the special value back

A ratio is 1/2, √2/2, √3/2, √3/3, 1 or √3, and you want the angle.

Sine: 1/2 is 30°, √2/2 is 45°, √3/2 is 60°.
Cosine runs the other way: 1/2 is 60°, √2/2 is 45°, √3/2 is 30°.
Tangent: √3/3 is 30°, 1 is 45°, √3 is 60°.

worked example

Example: In a right triangle, cos A = √3/2. What is angle A in degrees?

Cosine runs opposite to sine.
cos 30° = √3/2, so A is 30°.

Answer: 30

watch out for

Using the leg next to A for sine. Sine uses the opposite leg; cosine uses the leg next to A.
Mixing up 30° and 60°. sin 30° and cos 60° are both 1/2.
Dividing the hypotenuse by the sine instead of multiplying. A leg is always shorter than the hypotenuse.

practice

Trig ratios

worked example

In a right triangle, the leg opposite angle A is 3, the leg next to angle A is 4, and the hypotenuse is 5. What is tan A? Give a fraction.

Answer: 3/4

Tangent is opposite ÷ the leg next to A: 3/4.

Sides from special angles

worked example

A right triangle has hypotenuse 20 and a 30° angle. How long is the leg opposite the 30° angle?

Answer: 10

sin 30° = 1/2, so the leg opposite the angle is 20 × 1/2 = 10.

Angle from a ratio

worked example

In a right triangle, sin A = 1/2. What is angle A in degrees?

Answer: 30

sin 30° = 1/2, so A = 30°.