SOH-CAH-TOA (Sine, Cosine, Tangent)

In a right triangle, the three trig ratios compare pairs of sides relative to one angle.

θ = OH, θ = AH, θ = OA

From the angle theta: opposite is across, adjacent is beside, hypotenuse is the long slanted side.

What each symbol means

• θ: the angle you are looking at (not the right angle)
• O: the opposite side (across from the angle)
• A: the adjacent side (next to the angle, but not the hypotenuse)
• H: the hypotenuse (the longest side, opposite the right angle)

Trigonometry starts with three ratios that connect the angles of a right triangle to the lengths of its sides. The nine letters SOH-CAH-TOA are just a way to remember which sides go with which ratio.

Read it in three chunks. SOH: Sine is Opposite over Hypotenuse. CAH: Cosine is Adjacent over Hypotenuse. TOA: Tangent is Opposite over Adjacent.

Opposite and adjacent depend on the angle: The hypotenuse is always the longest side. But which side is "opposite" and which is "adjacent" depends on which angle you picked. Always name the sides from the angle you are working with.

A worked example

A right triangle has opposite side 3 and hypotenuse 5. Find θ.

1. Pick the ratio that uses opposite and hypotenuse: that is sine (SOH).
2. Write it: θ = OH.
3. Put in the numbers: θ = 35.
4. As a decimal: θ = 0.6.

Choosing the right ratio

You know the adjacent side (4) and the opposite side (3). Which ratio finds the angle?

1. You have opposite and adjacent, so use tangent (TOA).
2. Write it: θ = OA = 34.
3. To get the angle itself, you would use the inverse: θ = ^-1( 34), which is about 36.9 degrees.

• SOH: θ = OH.
• CAH: θ = AH.
• TOA: θ = OA.
• Name opposite and adjacent from the angle you are using; the hypotenuse never changes.

Why the ratios only depend on the angle

1. Take a right triangle with a certain angle, then imagine scaling it up to a bigger triangle with the same angles.
2. Every side gets multiplied by the same scale factor, so the shape stays identical, just larger.
3. When you form a ratio like opposite over hypotenuse, that shared scale factor cancels out (top and bottom both grow by the same amount).
4. So the ratio depends only on the angle, not on the triangle size. That is why sine, cosine, and tangent can be listed for each angle once and used for any right triangle.

Practice questions

Question 1

Opposite is 6, hypotenuse is 10. Find θ.

1. You have the opposite and the hypotenuse, so use sine (SOH).
2. Write the ratio: θ = OH.
3. Put in the numbers: θ = 610.
4. As a decimal, 610 = 0.6.

Answer: 0.6

Question 2

Adjacent is 8, hypotenuse is 10. Find θ.

1. You have the adjacent and the hypotenuse, so use cosine (CAH).
2. Write the ratio: θ = AH.
3. Put in the numbers: θ = 810.
4. As a decimal, 810 = 0.8.

Answer: 0.8

Question 3

Opposite is 5, adjacent is 12. Find θ.

1. You have the opposite and the adjacent, so use tangent (TOA).
2. Write the ratio: θ = OA.
3. Put in the numbers: θ = 512.
4. That fraction does not simplify, so the answer is 512.

Answer: 512

Question 4

You know the opposite side and the hypotenuse. Which ratio should you use?

1. Match the sides you have to the memory trick SOH-CAH-TOA.
2. SOH means Sine is Opposite over Hypotenuse.
3. Those are exactly the two sides you were given.
4. So the ratio to use is sine (SOH).

Answer: Sine (SOH)

Question 5

For a 3-4-5 right triangle, find θ where the adjacent is 4.

1. In a 3-4-5 triangle the hypotenuse is the longest side, which is 5.
2. The adjacent side is given as 4, so use cosine (CAH).
3. Write the ratio: θ = AH = 45.
4. As a decimal, 45 = 0.8.

Answer: 0.8