Chapter 14

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cos \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \csc \theta=-1 $$

Verify each identity. $$ \cos (A+B)=\cos A \cos B-\sin A \sin B $$

Express the first trigonometric function in terms of the second. $$ \sin \theta, \cos \theta $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(m \angle B=17.2^{\circ}, b=8.3\)

In \(\triangle A B C, m \angle A=40^{\circ}\) and \(m \angle B=30^{\circ} .\) Find each value to the nearest tenth. Find \(B C\) for \(A C=21.8 \mathrm{ft}\)

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \tan \frac{\theta}{2} $$

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \cot \theta=10 $$

Verify each identity. $$ \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} $$

In \(\triangle A B C, m \angle A=40^{\circ}\) and \(m \angle B=30^{\circ} .\) Find each value to the nearest tenth. Find \(A C\) for \(A B=81.2\) yd.

Express the first trigonometric function in terms of the second. $$ \tan \theta, \cos \theta $$

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(m \angle B=8.3^{\circ}, c=20\)

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cot \frac{\theta}{2} $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle T\) for \(r=11 \mathrm{ft}\)

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \csc \theta=3 $$

A 150 -ft pole casts a shadow 210 \(\mathrm{ft}\) long. Find the measure of the angle of elevation of the sun.

Verify each identity. $$ \sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=\sin x $$

Express the first trigonometric function in terms of the second. $$ \cot \theta, \sin \theta $$

In \(\triangle A B C, m \angle A=40^{\circ}\) and \(m \angle B=30^{\circ} .\) Find each value to the nearest tenth. Find \(B C\) for \(A B=5.9 \mathrm{cm}\)

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ 4 \sin 2 \theta-3 \cos \theta=0 $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle T\) for \(r=6.97 \mathrm{ft}\)

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \cot \theta=-10 $$

A transit is 330 \(\mathrm{ft}\) from the base of a building. The angles of elevation of the top and bottom of a flagpole situated on top of the building are \(55^{\circ}\) and \(53^{\circ}\) . Find the height of the flagpole.

Verify each identity. $$ \sin \left(\frac{3 \pi}{2}-x\right)=-\cos x $$

Measurement A vacant lot is in the shape of an isosceles triangle. It is between two streets that intersect at an \(85.9^{\circ}\) angle. Each of the sides of the lot that face these streets is 150 \(\mathrm{ft}\) long. Find the length of the third side, to the nearest foot.

Express the first trigonometric function in terms of the second. $$ \csc \theta, \cot \theta $$

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ 2 \sin 2 \theta-3 \sin \theta=0 $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle S\) for \(r=14 \mathrm{ft}\)

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \sec \theta=1 $$

An altitude inside a triangle forms angles of \(36^{\circ}\) and \(42^{\circ}\) with two of the sides. The altitude is 5 \(\mathrm{m}\) long. Find the area of the triangle.

Reasoning Show that the equation \(\sin (A+B)=\sin A+\sin B\) is not an identity by finding a counterexample, values for \(A\) and \(B\) for which the equation is false.

Sailing Buoys are located in the sea at points \(A, B,\) and \(C . \angle A C B\) is a right angle. \(A C=3.0 \mathrm{mi}, B C=4.0 \mathrm{mi},\) and \(A B=5.0 \mathrm{mi}\) A ship is located at point \(D\) on \(\overline{A B}\) so that \(m \angle A C D=30^{\circ} .\) How far is the ship from the buoy at point \(C ?\) Round your answer to the nearest tenth of a mile.

Express the first trigonometric function in terms of the second. $$ \cot \theta, \csc \theta $$

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ \sin 2 \theta \sin \theta=\cos \theta $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(r\) for \(m \angle R=35^{\circ}\)

Rewrite each expression as a trigonometric function of a single angle measure. $$ \sin 2 \theta \cos \theta+\cos 2 \theta \sin \theta $$

Writing Suppose you know the measures of all three angles of a triangle. Can you use the Law of Sines to find the lengths of the sides? Explain.

Express the first trigonometric function in terms of the second. $$ \sec \theta, \tan \theta $$

Use identities to write each equation in terms of the single angle \(\theta .\) Then solve the equation for \(0 \leq \theta<2 \pi .\) $$ \cos 2 \theta=-2 \cos ^{2} \theta $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle S\) for \(m \angle R=87^{\circ}\)

Rewrite each expression as a trigonometric function of a single angle measure. $$ \sin 3 \theta \cos 2 \theta+\cos 3 \theta \sin 2 \theta $$

Verify each identity. $$ \sin ^{2} \theta \tan ^{2} \theta=\tan ^{2} \theta-\sin ^{2} \theta $$

Simplify each expression. $$ 2 \cos ^{2} \theta-\cos 2 \theta $$

If \(\sin \theta=\frac{1}{2},\) describe a method you could use to find all the angles between \(0^{\circ}\) and \(360^{\circ}\) that satisfy this equation.

Find the complete solution in radians of each equation. $$ 2 \sin ^{2} \theta+\cos \theta-1=0 $$

Rewrite each expression as a trigonometric function of a single angle measure. $$ \cos 3 \theta \cos 4 \theta-\sin 3 \theta \sin 4 \theta $$

Verify each identity. $$ \sec \theta-\sin \theta \tan \theta=\cos \theta $$

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle R\) for \(m \angle S=70^{\circ}\)

In \(\triangle G D L, m \angle D=57^{\circ}, D L=10.1,\) and \(G L=9.4 .\) What is the best estimate for \(m \angle G ?\) \(\begin{array}{lllll}{\text { A. } 64^{\circ}} & {\text { B. } 51^{\circ}} & {\text { C. } 39^{\circ}} & {\text { D. } 26^{\circ}}\end{array}\)

Simplify each expression. $$ \sin ^{2} \frac{\theta}{2}-\cos ^{2} \frac{\theta}{2} $$