Q: A new homeowner has a triangular backyard with sides 53 ft and 42 ft and an included angle of $$135^ ext{o}. $$What is the area of the yard to the nearest square foot?

$$\frac{1}{2} \times 53 \times 42 \times \sin 135^ ext{o}$$

Q: Convert the rectangular coordinates $$(x, y) = (3, 4) to $$polar coordinates $$(r, \theta).$$

$$r = \sqrt{3^2$ + 4^2$},\ \theta = \tan$$^{-1}$$\left(\frac{4}{3}\right)$$

Q: What is the angle $$\theta$$ between two nonzero vectors $$\vec{u}$$ and $$\vec{v}$$ given by?

$$\cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}$$

Question 1

Which of the following is the Law of Sines for triangle $$ABC$$ with sides $$a$$, $$b$$, and $$c$$?

Accepted Answer

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

Question 2

A cruise ship is located N$$77^	ext{43}$$' E of Miami and N$$19^	ext{35}$$' E of Nassau. Miami is 178 mi N$$73^	ext{10}$$' W of Nassau. Which law would you use to find the distance from the ship to Nassau?

Accepted Answer

Law of Sines

Question 3

Given triangle $$ABC$$ with $$a = 10$$, $$A = 23^	ext{o}$$, and $$C = 131^	ext{o}$$, what is the correct setup to solve for side $$c$$ using the Law of Sines?

Accepted Answer

$$\frac{10}{\sin 23^	ext{o}} = \frac{c}{\sin 131^	ext{o}}$$

Question 4

Which scenario describes the ambiguous case (SSA) for solving triangles?

Accepted Answer

Two sides and an angle opposite one of them are known, possibly resulting in zero, one, or two solutions.

Question 5

A new homeowner has a triangular backyard with sides 53 ft and 42 ft and an included angle of $$135^	ext{o}. $$What is the area of the yard to the nearest square foot?

Accepted Answer

$$\frac{1}{2} 	imes 53 	imes 42 	imes \sin 135^	ext{o}$$

Question 6

Which formula is the Law of Cosines for side $$a in $$triangle $$ABC$$?

Accepted Answer

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

Question 7

Given the points $$A(-2,2)$$ and $$C(-2,5)$$, what is the component form of vector $$\vec{AC}$$?

Accepted Answer

$$\langle 0, 3 \rangle$$

Question 8

What is the magnitude of the vector $$\vec{v} = \langle 3, 4 \rangle$$?

Accepted Answer

$$\sqrt{3^2$ + 4^2$}$$

Question 9

Which of the following is the dot product of $$\vec{u} = \langle 2, 5 \rangle$$ and $$\vec{v} = \langle 3, 1 \rangle$$?

Accepted Answer

$$2 	imes 3 + 5 	imes 1$$

Question 10

If $$\vec{u} = \langle 1, 2 \rangle$$ and $$\vec{v} = \langle 1, 2 \rangle$$, are these vectors equivalent?

Accepted Answer

Yes, because their components are equal.

Question 11

Given a vector $$\vec{w} = \langle 3, 5 \rangle$$, what is the unit vector in the direction of $$\vec{w}$$?

Accepted Answer

$$\frac{\langle 3, 5 \rangle}{\sqrt{3^2$ + 5^2$}}$$

Question 12

Convert the rectangular coordinates $$(x, y) = (3, 4) to $$polar coordinates $$(r, 	heta).$$

Accepted Answer

$$r = \sqrt{3^2$ + 4^2$},\ 	heta = 	an$$^{-1}$$\left(\frac{4}{3}ight)$$

Question 13

Which formula expresses the area $$K of $$triangle $$ABC$$ using sides $$a$$ and $$b$$ and included angle $$C$$?

Accepted Answer

$$K = \frac{1}{2} ab \sin C$$

Question 14

If $$\vec{u} = \langle 2, 3 \rangle$$ and $$\vec{v} = \langle -1, 4 \rangle$$, what is $$\vec{u} + \vec{v}$$?

Accepted Answer

$$\langle 1, 7 \rangle$$

Question 15

What is the angle $$	heta$$ between two nonzero vectors $$\vec{u}$$ and $$\vec{v}$$ given by?

Accepted Answer

$$\cos 	heta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}$$

Question 16

Which of the following is the correct formula for multiplying two complex numbers in trigonometric form?

Accepted Answer

$$r_1$$ $$r_2$$ \left[ \cos(\$$theta_1$$ + \$$theta_2) + i$$ \sin(\$$theta_1$$ + \$$theta_2$$) \right]$$

Question 17

Given $$\vec{v} = \langle 2, 5 \rangle$$, express $$\vec{v} as a $$linear combination of unit vectors $$\vec{i}$$ and $$\vec{j}.$$

Accepted Answer

$$2\vec{i} + 5\vec{j}$$

Applications of Trigonometry: Laws, Complex Numbers, Polar Coordinates, and Vectors

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