Chapter 7

Each of the following problems refers to triangle \(A B C\). $$ \text { If } b=4.2 \mathrm{~m}, c=6.8 \mathrm{~m} \text {, and } A=116^{\circ} \text {, find } a \text {. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(3,-3)$$

Find all solutions to each of the following triangles: \(C=27^{\circ} 50^{\prime}, c=347 \mathrm{~m}, b=425 \mathrm{~m}\)

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=6 \mathbf{i}, \mathbf{V}=-8 \mathbf{j}\)

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=3.7 \mathrm{~m}, c=6.4 \mathrm{~m} \text {, and } B=33^{\circ} \text {, find } b \text {. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(5,-5)$$

Find all solutions to each of the following triangles: \(C=51^{\circ} 30^{r}, c=707 \mathrm{~m}, b=821 \mathrm{~m}\)

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=2 \mathrm{i}+5 \mathrm{j}, \mathbf{V}=5 \mathrm{i}+2 \mathrm{j}\)

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=38 \mathrm{~cm}, b=10 \mathrm{~cm} \text {, and } c=31 \mathrm{~cm} \text {, find the largest angle. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(-6,-4)$$

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=5 \mathbf{i}+3 \mathbf{j} . \mathbf{V}=-5 \mathbf{i}+3 \mathbf{j}\)

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=51 \mathrm{~cm}, b=24 \mathrm{~cm} \text {, and } c=41 \mathrm{~cm} \text {, find the largest angle. } $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in component form \(\langle a, b\rangle\). $$(-4,-6)$$

$$ \text { Solve each of the following triangles. } $$ $$ a=412 \mathrm{~m}, c=342 \mathrm{~m}, B=151.5^{\circ} $$

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=-4 \mathrm{i}-3 \mathrm{j}, \mathbf{V}=-\mathrm{i}-2 \mathrm{j}\)

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(2,5)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(A=42.5^{\circ}, B=71.4^{\circ}, a=210\) in.

Find all solutions to each of the following triangles: \(B=118^{\circ}, b=0.68 \mathrm{~cm}, a=0.92 \mathrm{~cm}\)

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=2 \mathrm{i}+9 \mathrm{j}, \mathrm{V}=-3 \mathrm{i}-\mathbf{j}\)

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(5,2)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(A=110.4^{\circ}, C=31.8^{\circ}, c=240\) i

Find all solutions to each of the following triangles: \(B=34^{\circ}, b=4.2 \mathrm{~cm}, a=4.2 \mathrm{~cm}\)

$$ \text { Solve each of the following triangles. } $$ $$ a=48 \mathrm{yd}, b=75 \mathrm{yd}, c=63 \mathrm{yd} $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(-3,6)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(A=43^{\circ} 30^{\circ}, C=120^{\circ} 30^{\circ}, a=3.48 \mathrm{ft}\)

Find all solutions to each of the following triangles: \(A=142^{\circ}, b=2.9 \mathrm{yd}, a=1.4 \mathrm{yd}\)

$$ \text { Solve each of the following triangles. } $$ $$ a=0.48 \mathrm{yd}, b=0.63 \mathrm{yd}, c=0.75 \mathrm{yd} $$

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=-11 \mathrm{i}+7 \mathbf{j}, \mathbf{V}=9 \mathrm{i}-5 \mathrm{j}\)

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(-6,3)$$

In triangle \(A B C, A=30^{\circ}, b=20 \mathrm{ft}\), and \(a=2 \mathrm{ft}\). Show that it is impossible to solve this triangle by using the law of sines to find \(\sin B\).

$$ \text { Solve each of the following triangles. } $$ $$ b=0.923 \mathrm{~km}, c=0.387 \mathrm{~km}, A=43^{\circ} 20^{\prime} $$

For each pair of vectors, find \(\mathbf{U} \cdot \mathbf{V}\). \(\mathbf{U}=5 \mathbf{i}-11 \mathbf{j}, \mathbf{V}=-20 \mathbf{i}+9 \mathbf{j}\)

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(4,-5)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=44\) in., \(b=66\) in., \(c=88\) in.

Find all solutions to each of the following triangles: \(C=26.8^{\circ}, c=36.8 \mathrm{~km}, b=36.8 \mathrm{~km}\)

In triangle \(A B C, A=40^{\circ}, b=19 \mathrm{ft}\), and \(a=18 \mathrm{ft}\). Use the law of sines to find \(\sin B\) and then give two possible values for \(B\).

$$ \text { Solve each of the following triangles. } $$ $$ b=63.4 \mathrm{~km}, c=75.2 \mathrm{~km}, A=124^{\circ} 40^{\prime} $$

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(5,-4)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=12\) in., \(b=23\) in., \(c=34\) in.

Find all solutions to each of the following triangles: \(C=83.4^{\circ}, c=51.1 \mathrm{~km}, b=94.2 \mathrm{~km}\)

$$ \text { Solve each of the following triangles. } $$ $$ a=4.38 \mathrm{ft}, b=3.79 \mathrm{ft}, c=5.22 \mathrm{ft} $$

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(\mathbf{U}=13 \mathbf{i}, \mathbf{V}=-6 \mathbf{j}\)

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=4.8 \mathrm{yd}, b=6.3 \mathrm{yd}, c=7.5 \mathrm{yd}\)

Distance A 51 -foot wire running from the top of a tent pole to the ground makes an angle of \(58^{\circ}\) with the ground. If the length of the tent pole is 44 feet, how far is it from the bottom of the tent pole to the point where the wire is fastened to the ground? (The tent pole is not necessarily perpendicular to the ground.)

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(U=-4 i, V=17 j\)

Draw the vector \(\mathbf{V}\) that goes from the origin to the given point. Then write \(\mathbf{V}\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$(-5,-1)$$

Each of the following problems refers to triangle \(A B C\). In each case, find the area of the triangle. Round to three significant digits. \(a=48 \mathrm{yd}, b=57 \mathrm{yd}, c=63 \mathrm{yd}\)

Distance A hot-air balloon is held at a constant altitude by two ropes that are anchored to the ground. One rope is 120 feet long and makes an angle of \(65^{\circ}\) with the ground. The other rope is 115 feet long. What is the distance between the points on the ground at which the two ropes are anchored?

$$ \text { Solve each of the following triangles. } $$ $$ \text { Use the law of cosines to show that, if } A=90^{\circ} \text {, then } a^{2}=b^{2}+c^{2} \text {. } $$

Find the angle \(\theta\) between the given vectors to the nearest tenth of a degree. \(\mathbf{U}=-3 \mathbf{i}+5 \mathbf{j}, \mathbf{V}=6 \mathbf{i}+3 \mathbf{j}\)