$$\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

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

$$\text{Area} = \frac{1}{2}(11.3)(14.7)\sin(84.1^\\circ)$$

Question 1

Given a triangle with $$a = 10.6$$, $$b = 12.3$$, and $$C = 29.4^\circ$$, which method should be used to solve for the remaining sides and angles?

Accepted Answer

Law of Cosines, since two sides and the included angle are given (SAS)

Question 2

A triangle has $$A = 39.7^\circ$$, $$B = 70.5^\circ$$, and $$C = 69.8^\circ. $$What can be concluded about the triangle?

Accepted Answer

There are infinitely many solutions since only the angles are given (AAA)

Question 3

Given $$a = 15.9$$, $$b = 12.3$$, and $$c = 14.9$$, which method should be used to solve for the angles of the triangle?

Accepted Answer

Law of Cosines, since all three sides are given (SSS)

Question 4

A triangle has $$A = 70.8^\circ$$, $$B = 46.9^\circ$$, and $$c = 14.9. $$What is the first step to solve for the remaining sides?

Accepted Answer

Find angle $$C by $$subtracting the sum of $$A$$ and $$B$$ from $$180^\circ$$

Question 5

Given $$A = 39.7^\circ$$, $$b = 31.4$$, and $$c = 10.5$$, which case does this triangle represent?

Accepted Answer

SSA (Side-Side-Angle), the ambiguous case

Question 6

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$\cos(2x) = \frac{\sqrt{3}}{2}.$$

Accepted Answer

$$x = \frac{\pi}{12}, \frac{11\pi}{12}, \frac{13\pi}{12}, \frac{23\pi}{12}$$

Question 7

What is the solution set for $$4\sin^2$ 	heta = 1 in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 8

Which formula gives the area of a triangle when two sides and the included angle are known (SAS)?

Accepted Answer

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

Question 9

Given $$a = 11.3$$, $$b = 14.7$$, and $$C = 84.1^\circ$$, what is the area of the triangle?

Accepted Answer

$$	ext{Area} = \frac{1}{2}(11.3)(14.7)\sin(84.1^\circ)$$

Question 10

Which of the following is the correct expansion for $$\sin(5x)\cos(7x) as a $$sum?

Accepted Answer

$$\frac{1}{2}[\sin(12x) + \sin(-2x)]$$

Question 11

Simplify $$	an\left(\frac{\pi}{4}ight)$$ using an identity.

Accepted Answer

$$1$$

Question 12

Verify the identity: $$\cot^2$ x - 	an^2$ x = \csc^2$ x - \sec^2$ x. $$Which of the following is a correct step in the proof?

Accepted Answer

Express $$\cot^2$ x$$ and $$	an^2$ x in $$terms of sine and cosine

Question 13

Given $$a = 7.1$$, $$b = 10.5$$, and $$C = 69.8^\circ$$, which method should be used to solve for the remaining sides and angles?

Accepted Answer

Law of Sines, since two sides and a non-included angle are given (SSA)

Question 14

If $$a = 12.3$$, $$b = 14.9$$, and $$c = 15.9$$, what is the semi-perimeter $$s of $$the triangle?

Accepted Answer

$$s = \frac{12.3 + 14.9 + 15.9}{2}$$

Question 15

Which of the following is the correct product-to-sum formula for $$\sin(3x) - \sin(2x)$$?

Accepted Answer

$$2\sin\left(\frac{3x - 2x}{2}\right)\cos\left(\frac{3x + 2x}{2}\right)$$

Question 16

A triangle has $$A = 39.7^\circ$$, $$B = 70.5^\circ$$, and $$c = 10.5. $$What is the first step to solve for the remaining sides?

Accepted Answer

Find angle $$C by $$subtracting the sum of $$A$$ and $$B$$ from $$180^\circ$$

Question 17

Given $$A = 39.7^\circ$$, $$b = 22.2$$, and $$c = 10.5$$, which case does this triangle represent?

Accepted Answer

SSA (Side-Side-Angle), the ambiguous case

Solving Triangles and Trigonometric Equations in Precalculus

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