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Combination of SohCahToa questions

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SOHCAHTOA Combination Questions

SOHCAHTOA is a mnemonic for the three trig ratios: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent. Combination questions use more than one ratio in the same problem. Learn to choose the right ratio and work through a two-step worked example.

The SOHCAHTOA mnemonic

SOHCAHTOA is a memory aid for the three basic trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Combination questions ask you to use more than one of these ratios in the same problem.

SOHCAHTOA worked triangle A right triangle with the reference angle 40 degrees, adjacent side 20 meters. Using tan 40 degrees equals opposite over adjacent, the opposite side is about 16.8 meters. Using cos 40 degrees, the hypotenuse is about 26.1 meters. 40° adjacent = 20 m opposite ≈ 16.8 m hypotenuse ≈ 26.1 m
Given the angle and one side, tangent finds the opposite side, then cosine finds the hypotenuse.

Worked example

A right triangle has a 40° angle and an adjacent side of 20 m. Find the opposite side and the hypotenuse.

Step 1 — find the opposite side (TOA): tan 40° = opposite / 20, so opposite = 20 · tan 40° ≈ 16.8 m.

Step 2 — find the hypotenuse (CAH): cos 40° = 20 / hypotenuse, so hypotenuse = 20 / cos 40° ≈ 26.1 m.

Choosing the right ratio

The key skill in a combination question is picking the correct ratio for what you know and what you need. If you know the angle and the adjacent side but want the opposite side, tangent connects them directly — there's no need to find the hypotenuse first. This builds on the individual ratios: the tangent ratio, the sine ratio, and the cosine ratio.

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