Chapter 5

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle\sqrt{3},-1\rangle $$

Use the Law of Cosines to solve the triangle. $$ \gamma=65^{\circ}, a=5, b=8 $$

A building casts a shadow \(20 \mathrm{~m}\) long. If the angle from the tip of the shadow to a point on top of the building is \(69^{\circ}\), how high is the building?

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle 4,-4\rangle $$

Use the Law of Cosines to solve the triangle. $$ \beta=48^{\circ}, a=7, c=6 $$

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle 5,0\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=8, b=10, c=7 $$

A 50 -ft tower is located on the edge of a river. The angle of elevation between the opposite bank and the top of the tower is \(37^{\circ} .\) How wide is the river?

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \langle-2,2 \sqrt{3}\rangle $$

Use the Law of Cosines to solve the triangle. $$ \gamma=31.5^{\circ}, a=4, b=8 $$

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ -4 \mathbf{i}+4 \sqrt{3} \mathbf{j} $$

An observer on the roof of building \(A\) measures a \(27^{\circ}\) angle of depression between the horizontal and the base of building \(B\). The angle of elevation from the same point to the roof of the second building is \(41.42^{\circ} .\) What is the height of building \(B\) if the height of building \(A\) is \(150 \mathrm{ft}\) ? Assume buildings \(A\) and \(B\) are on the same horizontal plane.

Use the Law of Sines to solve the triangle. $$ \alpha=80^{\circ}, \beta=20^{\circ}, b=7 $$

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ \mathbf{i}-\mathbf{j} $$

Use the Law of Cosines to solve the triangle. $$ a=7, b=9, c=4 $$

Use the Law of Sines to solve the triangle. $$ \alpha=80^{\circ}, \beta=20^{\circ}, b=7 $$

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ -10 \mathbf{i}+10 \mathbf{j} $$

Use the Law of Cosines to solve the triangle. $$ a=11, b=9 \cdot 5, c=8.2 $$

The top of a 20 -ft ladder is leaning against the edge of the roof of a house. If the angle of inclination of the ladder from the horizontal is \(51^{\circ}\), what is the approximate height of the house and how far is the bottom of the ladder from the base of the house?

Use the Law of Sines to solve the triangle. $$ \beta=37^{\circ}, \gamma=51^{\circ}, a=5 $$

Sketch the given vector. Find the magnitude and the smallest positive direction angle of each vector. $$ -3 \mathbf{j} $$

Use the Law of Cosines to solve the triangle. $$ \alpha=162^{\circ}, b=11, c=8 $$

An airplane flying horizontally at an altitude of \(25,000 \mathrm{ft}\) approaches a radar station located on a 2000 -ft-high hill. At one instant in time, the angle between the radar dish pointed at the plane and the horizontal is \(57^{\circ} .\) What is the straight-line distance in miles between the airplane and the radar station at that particular instant?

Use the Law of Sines to solve the triangle. $$ \alpha=30^{\circ}, \gamma=75^{\circ}, a=6 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle 2,3\rangle, \mathbf{v}=\langle 1,-1\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=5, b=7, c=10 $$

A 5 -mi straight segment of a road climbs a 4000 ft hill. Determine the angle that the road makes with the horizontal.

Use the Law of Sines to solve the triangle. $$ \beta=72^{\circ}, b=12, c=6 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle 10,2\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=6, b=5, c=7 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle-4,2\rangle, \mathbf{v}=\langle 4,1\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=3, b=4, c=5 $$

Use the Law of Sines to solve the triangle. $$ \gamma=62^{\circ}, b=7, c=4 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle-1,-5\rangle, \mathbf{v}=\langle 8,7\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=5, b=12, c=13 $$

Use the Law of Sines to solve the triangle. $$ \beta=110^{\circ}, \gamma=25^{\circ}, a=14 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle-5,-7\rangle, \mathbf{v}=\left\langle\frac{1}{2},-\frac{1}{4}\right\rangle $$

Use the Law of Cosines to solve the triangle. $$ a=6, b=8, c=12 $$

Use the Law of Sines to solve the triangle. $$ \gamma=15^{\circ}, a=8, c=5 $$

Find \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},-3 \mathbf{u},\) and \(3 \mathbf{u}-\) \(4 \mathbf{v}\). $$ \mathbf{u}=\langle 0.1,0.2\rangle, \mathbf{v}=\langle-0.3,0.4\rangle $$

Use the Law of Cosines to solve the triangle. $$ \beta=130^{\circ}, a=4, c=7 $$

From an observation site \(1000 \mathrm{ft}\) from the base of Mt. Rushmore the angle of elevation to the top of the sculpted head of George Washington is measured to be \(80.05^{\circ},\) whereas the angle of elevation to the bottom of his head is \(79.946^{\circ}\). Determine the height of George Washington's head.

Use the Law of Sines to solve the triangle. $$ \alpha=55^{\circ}, a=20, c=18 $$

Find \(\mathbf{u}-4 \mathbf{v}\) and \(2 \mathbf{u}+5 \mathbf{v}\). $$ \mathbf{u}=\mathbf{i}-2 \mathbf{j}, \mathbf{v}=8 \mathbf{i}+3 \mathbf{j} $$

Use the Law of Sines to solve the triangle. $$ \gamma=150^{\circ}, b=7, c=5 $$

Find \(\mathbf{u}-4 \mathbf{v}\) and \(2 \mathbf{u}+5 \mathbf{v}\). $$ \mathbf{u}=\mathbf{j}, \mathbf{v}=4 \mathbf{i}-\mathbf{j} $$

Use the Law of Cosines to solve the triangle. $$ \beta=100^{\circ}, a=22.3, b=16.1 $$

The height of a gnomon (pin) of a sundial is 4 in. If it casts a 6 -in. shadow, what is the angle of elevation of the Sun?

Use the Law of Sines to solve the triangle. $$ \alpha=35^{\circ}, a=9, b=12 $$

Find \(\mathbf{u}-4 \mathbf{v}\) and \(2 \mathbf{u}+5 \mathbf{v}\). $$ \mathbf{u}=\frac{1}{2} \mathbf{i}-\frac{3}{2} \mathbf{j}, \mathbf{v}=2 \mathbf{i} $$