7. Non-Right Triangles

Given a right triangle with sides $a$, $b$, and hypotenuse $c$, which equation can be used to solve for $c$ using the Law of Cosines?

In triangle $BCD$, which is isosceles with $BC$ congruent to $BD$, what is the measure of angle $CBD$ if angle $C$ is $20°$? Choose the correct answer.

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

In triangle $\mathrm{ABC}$, if side $\mathrm{AB}$ has length $b$, side $\mathrm{BC}$ has length $a$, and the angle at vertex $C$ is $\gamma$, what is the length of side $\mathrm{AC}$ according to the Law of Cosines?

Solve each triangle. Approximate values to the nearest tenth.

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Solve each triangle. See Examples 2 and 3.

C = 45.6°, b = 8.94 m, a = 7.23 m

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.

$\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$

A surveyor wishes to find the distance across a river while standing on a small island. If she measures distances of $a=30m$ to one shore, $c=60m$ to the opposite shore, and an angle of $B=100\(\degree\)$ between the two shores, find the distance between the two shores.

Solve each triangle. See Examples 2 and 3.

a = 9.3 cm, b = 5.7 cm, c = 8.2 cm

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.