ACT — mathHard1 min read

Law of Sines and Cosines

Apply the law of sines and law of cosines to non-right triangles for the ACT.

Tutor AI Team2026-02-25T02:04:13.188Z

The law of sines and law of cosines extend trigonometry to non-right triangles. These appear on harder ACT questions.

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Use when: angle-side pair + one more piece.

$c^2 = a^2 + b^2 - 2ab\cos C$

Use when: SAS (two sides + included angle) or SSS (three sides).

$\text{Area} = \frac{1}{2}ab\sin C$

Problem

$A = 40°$ , $a = 8$ , $B = 60°$ . Find $b$ . $\frac{8}{\sin 40°} = \frac{b}{\sin 60°}$ → $b = \frac{8\sin 60°}{\sin 40°} \approx 10.78$ .

Solution

Problem

$a = 5$ , $b = 7$ , $C = 50°$ . Find $c$ . $c^2 = 25 + 49 - 70\cos 50° = 74 - 44.99 = 29.01$ . $c \approx 5.39$ .

Solution

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#act#math#trigonometry#law-of-sines#law-of-cosines

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