SOH CAH TOA and when to use each ratio

TL;DR:

• SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

• Opposite and adjacent depend on which angle you choose; the hypotenuse never changes.

• To find an angle, use the inverse function: sin⁻¹, cos⁻¹ or tan⁻¹.

A 5 m ladder rests against a wall with its foot 1.5 m from the wall. What angle does it make with the ground? Pythagoras gives lengths; this question needs angles. The three ratios called sine, cosine and tangent connect the two.

Why a ratio depends only on the angle

Take any right triangle with a 30° angle. Scale it up or down, and the sides change, but the ratio of the side opposite 30° to the hypotenuse stays the same, because the triangles are similar. Sine, cosine and tangent are those fixed ratios, tabulated for each angle.

Naming the sides from the angle

Pick the angle you care about, call it θ. The hypotenuse is the side across from the right angle. The opposite side is across from θ. The adjacent side is the one that touches θ and is not the hypotenuse. If you switch to the other acute angle, opposite and adjacent swap, but the hypotenuse stays.

Worked example: the angle of a ladder against a wall

The ladder is the hypotenuse, 5 m. The foot is 1.5 m from the wall, which is along the ground. Let θ be the angle between ladder and ground.

1. Label from θ. The ground distance 1.5 touches θ, so it is the adjacent side. The ladder is the hypotenuse, 5.

2. Pick the ratio that uses adjacent and hypotenuse: cosine, since CAH.

3. Write: cos θ = 1.5 / 5 = 0.3.

4. Invert: θ = cos⁻¹(0.3) ≈ 72.5°.

Now check with another ratio. The height on the wall is √(5² − 1.5²) = √(25 − 2.25) = √22.75 ≈ 4.77 m. Then sin θ = 4.77 / 5 ≈ 0.954, and sin 72.5° ≈ 0.954. Also tan θ = 4.77 / 1.5 ≈ 3.18, which matches tan 72.5° ≈ 3.18. All three ratios agree.

Finding a side instead of an angle

A ramp makes a 35° angle with the ground and is 10 m long. How high does it rise? The rise is opposite the angle and the ramp is the hypotenuse, so use sine: opp = 10 × sin 35°. Since sin 35° ≈ 0.5736, the rise is about 5.74 m. The run along the ground is 10 × cos 35° ≈ 10 × 0.8192 = 8.19 m. Check with Pythagoras: 5.74² + 8.19² ≈ 32.9 + 67.1 = 100, which is 10².

Try two more yourself

A kite string is 40 m long and makes a 50° angle with the ground. The height is opposite the angle and the string is the hypotenuse, so use sine: 40 × sin 50° ≈ 40 × 0.7660 = 30.6 m. Check by finding the horizontal distance, 40 × cos 50° ≈ 40 × 0.6428 = 25.7 m, and testing 30.6² + 25.7² ≈ 936 + 661 = 1597, close to 40² = 1600.

A tree casts a 12 m shadow when the sun is 40° above the horizon. The shadow is adjacent to the angle and the tree is opposite, so use tangent: 12 × tan 40° ≈ 12 × 0.8391 = 10.1 m. The tree is about 10 m tall.

Angles of elevation

Word problems often say "angle of elevation", which is the angle measured upward from the horizontal to your line of sight. It is the same θ as in the triangle, with the horizontal as the adjacent side and the height as the opposite side. Looking up at a tower from 30 m away at an elevation of 60°, the height is 30 × tan 60° = 30 × √3 ≈ 30 × 1.732 = 52 m, plus your eye height if it matters. Always draw the triangle first, even a rough one, and mark the angle before you choose a ratio.

Which ratio do I use?

• Know the angle, want a side, or know two sides, want an angle? Write down which two sides are involved.

• Opposite and hypotenuse: sine.

• Adjacent and hypotenuse: cosine.

• Opposite and adjacent: tangent.

• If all three sides are known, any ratio works, which is a good way to check.

How to check your answer

• Calculator mode. Make sure the calculator is in degrees when the problem uses degrees. sin 30 should give 0.5.

• Size of the result. The hypotenuse must be the longest side. Sine and cosine of an acute angle must be between 0 and 1.

• Pythagoras. Square the sides and test the sum.

• Reasonableness. A steep ladder should give a large angle. 72.5° is steep, and the foot is close to the wall.

Common mistakes

• Labelling opposite and adjacent from the wrong angle.

• Using the right angle as the reference angle.

• Leaving the calculator in radian mode. Then sin 30 shows about −0.988.

• Forgetting the inverse function when finding an angle: cos θ = 0.3 does not mean θ = 0.3.

• Using tangent when the hypotenuse is involved.

Where you will use it next

The ratios extend beyond right triangles to the unit circle, covered in the unit circle without memorizing a table. They also rest on the similar-triangle argument from similar triangles: measuring a tree with its shadow. Engineering uses them for slopes, forces on ramps and heights from a distance.

Frequently asked questions

Why is sine of an angle never greater than 1?

The opposite side of a right triangle is always shorter than the hypotenuse, so the ratio is below 1. Angles bigger than 90° have a different story, covered with the unit circle.

Is tangent the same as slope?

Yes, in effect. A line rising at an angle θ above horizontal has slope tan θ.