Chapter 14

If the sine of an angle of a triangle is known, is it possible to determine the measure of the angle? Explain why or why not.

Explain how the Law of Cosines can be used to show that \(4,7,\) and 12 cannot be the measures of the sides of a triangle.

Rosa found the area of parallelogram \(A B C D\) by using \((A B)(B C)(\sin B) .\) Riley found the area of parallelogram \(A B C D\) by using \((A B)(B C)(\sin A) .\) Explain why Rosa and Riley both got the correct answer.

Explain why an angle of depression is always congruent to an angle of elevation.

If the cosine of an angle of a triangle is known, is it possible to determine the measure of the angle? Explain why or why not.

Show that if \(\angle C\) is an obtuse angle, \(a^{2}+b^{2}

Explain why, when the measures of two sides and an obtuse angle opposite one of them are given, it is never possible to construct two different triangles.

Explain the relationship between the Law of Cosines and the Pythagorean Theorem.

Jessica said that the area of rhombus \(P Q R S\) is \((P Q)^{2}(\sin P) .\) Do you agree with Jessica? Explain why or why not.

In \(\triangle A B C,\) if \(a=9, \mathrm{m} \angle A=\frac{\pi}{3},\) and \(\mathrm{m} \angle B=\frac{\pi}{4},\) find the exact value of \(b\) in simplest form.

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=8, b=10, \mathrm{m} \angle A=20 $$

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ b=3, c=8, \sin A=\frac{1}{4} $$

In \(\triangle M A R,\) express \(m^{2}\) in terms of \(a, r,\) and \(\cos M\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=4, \theta=45^{\circ}\)

In \(\triangle A B C,\) if \(a=24, \mathrm{m} \angle A=\frac{\pi}{6},\) and \(\mathrm{m} \angle B=\frac{\pi}{2},\) find the exact value of \(b\) in simplest form.

In \(\triangle P Q R,\) express cos \(Q\) in terms of \(p, q,\) and \(r\)

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=5, b=10, \mathrm{m} \angle A=30 $$

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ a=12, c=15, \sin B=\frac{1}{3} $$

In \(\triangle N O P,\) express \(p^{2}\) in terms of \(n, o,\) and \(\cos P\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=2, \theta=30^{\circ}\)

In \(\triangle A B C,\) if \(c=12, \mathrm{m} \angle C=\frac{2 \pi}{3},\) and \(\mathrm{m} \angle B=\frac{\pi}{6},\) find the exact value of \(b\) in simplest form.

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ b=12, c=10, \mathrm{m} \angle B=49 $$

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ b=9, c=16, \sin A=\frac{5}{6} $$

In \(\triangle A B C,\) if \(a=3, b=5,\) and \(\cos C=\frac{1}{5},\) find the exact value of \(c\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=6, \theta=90^{\circ}\)

In \(\triangle A B C,\) if \(b=8, \mathrm{m} \angle A=\frac{\pi}{3},\) and \(\mathrm{m} \angle C=\frac{5 \pi}{12},\) find the exact value of \(a\) in simplest form.

In \(\triangle X Y Z,\) if \(x=1, y=2,\) and \(z=\sqrt{5},\) find the exact value of \(\cos Z\)

In \(\triangle D E F,\) if \(e=8, f=3,\) and \(\cos D=\frac{3}{4},\) find the exact value of \(d\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=8, \theta=120^{\circ}\)

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle A B C, a=4, b=6, c=8\)

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ c=18, b=10, \mathrm{m} \angle C=120 $$

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ b=7, c=8, \sin A=\frac{3}{5} $$

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=15, \theta=135^{\circ}\)

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle A B C, a=12, b=8, c=8\)

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=9, c=10, \mathrm{m} \angle C=150 $$

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ a=10, c=8, \sin B=\frac{3}{10} $$

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle A B C, b=4, c=4,\) and \(\mathrm{m} \angle A=\frac{\pi}{3}\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=0.5, \theta=180^{\circ}\)

In \(\triangle D E F, \mathrm{m} \angle D=47, \mathrm{m} \angle E=84,\) and \(d=17.3 .\) Find \(e\) to the nearest tenth.

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle D E F, d=15, e=12, f=8\)

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle A B C, b=14.6, c=12.8, \mathrm{m} \angle A=56\)

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle P Q R, p=6, q=\sqrt{2},\) and \(\mathrm{m} \angle R=\frac{\pi}{4}\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=9, \theta=150^{\circ}\)

In \(\triangle D E F, \mathrm{m} \angle D=56, \mathrm{m} \angle E=44,\) and \(d=37.5 .\) Find \(e\) to the nearest tenth.

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ D E=24, E F=18, \mathrm{m} \angle D=15 $$

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle A B C, a=326, c=157, \mathrm{m} \angle B=72\)

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle D E F, d=\sqrt{3}, e=5,\) and \(\mathrm{m} \angle F=\frac{\pi}{6}\)

Write in simplest radical form the coordinates of each point \(A\) if \(A\) is on the terminal side of an angle in standard position whose degree measure is \(\theta .\) \(O A=25, \theta=210^{\circ}\)

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle A B C, a=15, b=18,\) and \(\mathrm{m} \angle C=60\)

In \(\triangle L M N, \mathrm{m} \angle M=112, \mathrm{m} \angle N=54,\) and \(m=51.0 .\) Find \(n\) to the nearest tenth.