Right triangles as well as conforming to the Pythagorean theorem, conform to a set of rules about how the angle of one corner in the triangle corresponds to the lengths of the sides opposite it, adjacent to it, and the length of the hypotenuse (Figure 1). These functions also use the trigonometric functions sine, cosine and tangent. Collectively, these rules can be called SOHCAHTOA, which is a helpful mnemonic to remember.

Figure 1: A right triangle and the application of the SOHCAHTOA rules. From the perspective of any one of the non-right triangles in the triangle, there are three sides of the triangle. The hypotenuse is the longest side, which is opposite to the right triangle. The adjacent side, which is the side that is directly next to the perspective angle but is not the hypotenuse. And the opposite side, which is the side directly opposite from the perspective angle.

The SOHCAHTOA rules spell out SOH (Eq. 1), CAH (Eq. 2) and TOA (Eq. 3), which are three separate mathematical formulae, describing how the trigonometric functions applied to the angle of one corner relate to the sides of the triangle.

Eq. 1: \(\sin\theta = {o \over h}\)

Eq. 2: \(\cos\theta = {a \over h}\)

Eq. 3: \(\tan\theta = {o \over a}\)

Any two sides of the right triangle are related to one of the trigonometric functions, and all three of these trigonometric functions have some relation to two sides of the triangle (Figure 2)

Figure 2: A right triangle and the application of the SOHCAHTOA rules. From the perspective of any one of the non-right triangles in the triangle, there are three sides of the triangle. The hypotenuse is the longest side, which is opposite to the right triangle. The adjacent side, which is the side that is directly next to the perspective angle but is not the hypotenuse. And the opposite side, which is the side directly opposite from the perspective angle.

As there are many three values that can be derived using each of the three laws in this collection of laws, only one of each is going to be shown.

Using SOH to Find Hypotenuse

If we are given a triangle for which we know the angle of a corner (for example 35°), along with the length of the side opposite that angle (for example 43cm), we can determine the length of the hypotenuse, without needing to use the Pythagorean theorem or know the length of the adjacent side. To do this, we can rearrange the SOH law, as we have the angle of a corner, and the length of the opposite side, but are looking for the length of the hypotenuse. This gives (Eq. 4).

Eq. 4: \(h = {o \over \sin\theta}\)

If we have an angle of 35°, we can find the sine (Eq. 5). We can divide this by the length of an opposite side (43cm) to return the length of the hypotenuse (Eq. 6), which is 73.97cm.

Eq. 5: \(\sin35 = 0.574\)

Eq. 6: \(h = {43 \over 0.574} = 73.97\)

Using CAH to Find Adjacent

If we are given the angle of a corner (for example 63°), along with the length of its hypotenuse (for example 34cm), we can use the CAH law to find the length of the side adjacent to the angle. To do this we can rearrange the CAH law to find adjacent, which is by multiplying the cosine of the angle by the length of the hypotenuse (Eq. 7)

Eq. 7: \(a = \cos\theta\times h\)

After finding the cosine of 63 (Eq. 8), we can multiply this by the length 34 to give the length of the adjacent side of 15.44cm (Eq. 9).

Eq. 8: \(\cos63 = 0.454\)

Eq. 9: \(a = 0.454\times 34 = 15.44\)

TOA to Find Angle

If we know the length of the sides opposite and adjacent to a right triangle, for example 63cm and 42cm respectively we can use these to find the angle between the adjacent side and the hypotenuse. To do this we rearrange the TOA law to find the angle. Because the TOA law relates to the tangent of the angle and not the angle, we have to use the inverse tangent function, tan^-1 (Eq. 10).

Eq. 10: \(\theta = \sin^{-1}{o \over a}\)

We can then divide our opposite by our adjacent sides to find the tangent of our angle (Eq. 11). Although like other trigonometric functions, an infinite number of different angles have the same tangent value, only one of these angles will be a valid angle for a right triangle, and this is the angle returned by the inverse sign function on a scientific calculator. This equation thus returns the angle of interest between the adjacent side and the hypotenuse (Eq. 12), which is 56.31°.

Eq. 11: \({63 \over 42} = \tan\theta\)

Eq. 12: \(\theta = \tan^{-1} 1.5 = 56.31\)

SOHCAHTOA is a useful set of rules to for right angled triangles, and other shapes that contain right triangles. Also remember that the opposite and adjacent sides are interchangeable, as long as you also change the angle that you are using, as the adjacent and opposite sides are named with respect to this angle. Also remember that these two angles which are not the right angle always add up to 90°, as all the angles of triangles add up to 180°, and 90° of these angles are taken up by the right angle in the triangle.