Intro to SOHCAHTOA

Trigonometry

Lesson by Mike McGarry

Magoosh Expert

Lesson by Mike McGarry

Magoosh Expert

Summary

SOHCAHTOA comes directly from similar right triangles: for any fixed acute angle, the sine, cosine, and tangent ratios stay constant no matter how large or small the triangle is. On the ACT, these ratios help you identify opposite, adjacent, and hypotenuse correctly, solve for missing sides, find areas, and apply the key identity cosine squared theta plus sine squared theta equals one.

Similar triangles form the foundation of trigonometry because any two right triangles sharing the same acute angle must be similar, which makes corresponding side ratios constant.
Each side is named relative to a chosen angle: the hypotenuse is opposite the right angle, the adjacent side touches the chosen acute angle, and the opposite side lies across from it.
SOHCAHTOA encodes the three core ratios: sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent; calculator work must be done in degree mode for these ACT-style problems.
A right triangle with one acute angle and one known side can be solved by setting up the appropriate trig ratio and isolating the unknown length, as shown with examples using sine and tangent.
ACT problems often combine trig with geometry skills such as recognizing a 3-4-5 triangle, finding tangent from the correct angle’s perspective, and using height from a trig ratio to compute area.
The general angle theta allows the formulas to be written for any acute angle in a right triangle, and the Pythagorean theorem leads to the identity cosine squared theta plus sine squared theta equals one.
Multi-step word problems require choosing the correct ratio first and then interpreting the diagram carefully, such as using cosine to find a full segment and subtracting a known piece to get the actual change in height.

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