Calculate a missing side or angle of a right triangle from the sides and angle you already know, using the SOHCAHTOA trigonometric ratios.
SOHCAHTOA Formula
SOHCAHTOA is a memory aid for the three trigonometric ratios that relate an acute angle of a right triangle to its sides. Each ratio uses two of the three sides: the side opposite the angle, the side adjacent to the angle, and the hypotenuse.
sin(theta) = Opposite / Hypotenuse
cos(theta) = Adjacent / Hypotenuse
tan(theta) = Opposite / Adjacent
Where:
- theta = the known acute angle of the right triangle
- Opposite = the side directly across from the angle theta
- Adjacent = the side next to the angle theta that is not the hypotenuse
- Hypotenuse = the longest side, opposite the right angle
When you solve for a side, the calculator rearranges the matching ratio. For example, the opposite side equals Hypotenuse times sin(theta), and the adjacent side equals Hypotenuse times cos(theta). When you solve for the angle, it uses the inverse functions on the two sides you supply:
theta = arcsin(Opposite / Hypotenuse)
theta = arccos(Adjacent / Hypotenuse)
theta = arctan(Opposite / Adjacent)
The “Solve for” selector chooses which value you want to find, and the method choice tells the calculator which two known values you have. Pick degrees or radians for the angle, and a length unit so the side results are labeled. SOHCAHTOA applies only to right triangles, so the angle must be greater than 0 degrees and less than 90 degrees.
Trig Ratios at Common Angles
These exact values appear often in right-triangle problems and are useful for checking a result quickly.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.500 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1 |
| 60° | 0.866 | 0.500 | 1.732 |
| 90° | 1 | 0 | undefined |
To decide which ratio to use, match the two sides involved in your problem to the row below.
| Sides involved | Ratio to use | Mnemonic |
|---|---|---|
| Opposite and Hypotenuse | sine | SOH |
| Adjacent and Hypotenuse | cosine | CAH |
| Opposite and Adjacent | tangent | TOA |
Example Problems
Example 1: Find a side. A right triangle has an acute angle of 30 degrees and a hypotenuse of 10. To find the opposite side, use sine: Opposite = Hypotenuse times sin(theta) = 10 times sin(30°) = 10 times 0.5 = 5. The opposite side is 5.
Example 2: Find an angle. A right triangle has an opposite side of 3 and an adjacent side of 4. Because you know the opposite and adjacent sides, use the tangent: theta = arctan(3 / 4) = arctan(0.75) = 36.87 degrees.
Frequently Asked Questions
Does SOHCAHTOA work on any triangle? No. SOHCAHTOA is defined only for right triangles, where one angle is exactly 90 degrees. For triangles without a right angle, use the law of sines or the law of cosines instead.
How do I know which side is opposite and which is adjacent? The labels are always relative to the acute angle you are working with. The opposite side is the one that does not touch the angle, the hypotenuse is the longest side across from the right angle, and the adjacent side is the remaining side that touches the angle.
Why does my answer change between degrees and radians? The trig functions interpret the angle value according to the unit you select. An angle of 30 in degrees is not the same as 30 in radians, so set the angle unit to match your problem before you calculate.
