Back to Exercise: Apply Law of Sines and Cosines

Exercises: Apply the Law of Sines and the Law of Cosines

Grade 10·22 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-srt-d-11

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Recall / Warm-Up

Which formula correctly states the Law of Sines for triangle $ABC$ ?

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

$a^2 = b^2 + c^2 - 2bc\cos A$

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

$a = b\sin A + c\cos A$

A triangle has two sides and the included angle known (SAS). Which law should you use first to find the missing side?

Law of Sines, because you have two ratios to compare

Law of Cosines, because the known angle is between the two known sides

Pythagorean Theorem, because any triangle can be split into right triangles

Either law — they always give the same answer

A bearing of 150° points in which direction relative to north?

60° west of south

30° east of south

150° west of north

30° west of north

Fluency Practice

$Triangle ABC with angle A = 35 degrees, angle B = 85 degrees, and side a = 10, with side b to be found$

In triangle $ABC$ , angle $A = 35\degree$ , angle $B = 85\degree$ , and side $a = 10$ . Use $\sin(35\degree) \approx 0.574$ and $\sin(85\degree) \approx 0.996$ . Find side $b$ . Round to the nearest tenth.

In triangle $ABC$ , angle $A = 50\degree$ , angle $C = 70\degree$ , and side $c = 20$ . Use $\sin(50\degree) \approx 0.766$ and $\sin(70\degree) \approx 0.940$ . Find side $a$ . Round to the nearest tenth.

In triangle $ABC$ , angle $A = 30\degree$ , side $a = 8$ , and side $b = 12$ . Compute $\sin B = \dfrac{b \sin A}{a} = \dfrac{12 \times 0.5}{8} = 0.75$ . How many valid triangles exist with these measurements?

No valid triangles — $\sin B > 1$

Exactly one triangle — the acute triangle only

Exactly two triangles — $B \approx 48.6\degree$ or $B \approx 131.4\degree$

Exactly one triangle — the obtuse triangle only

$Triangle ABC with sides a = 8 and b = 11 and included angle C = 50 degrees, with side c to be found$

In triangle $ABC$ , side $a = 8$ , side $b = 11$ , and angle $C = 50\degree$ . Use $\cos(50\degree) \approx 0.643$ . Find side $c$ . Round to the nearest tenth.

In triangle $ABC$ , all three sides are known: $a = 5$ , $b = 7$ , $c = 9$ . Use the Law of Cosines to find angle $C$ (the largest angle, opposite side $c$ ). Use $\cos^{-1}(-0.100) \approx 95.7\degree$ . Round to the nearest tenth of a degree.

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