Chapter 6

Find the exact value of the expression. $$ \sin \left(\cos ^{-1} \frac{3}{5}\right) $$

\(19-28\) . Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=26, \quad c=15, \quad \angle C=29^{\circ} $$

Find the exact value of the trigonometric function. $$ \cos \left(-\frac{7 \pi}{3}\right) $$

Evaluate the expression without using a calculator. $$ \sin 30^{\circ} \cos 60^{\circ}+\sin 60^{\circ} \cos 30^{\circ} $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ 50^{\circ} $$

Find the exact value of the expression. $$ \tan \left(\sin ^{-1} \frac{4}{5}\right) $$

Find the exact value of the trigonometric function. $$ \tan \frac{5 \pi}{6} $$

Evaluate the expression without using a calculator. $$ \left(\sin 60^{\circ}\right)^{2}+\left(\cos 60^{\circ}\right)^{2} $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ 135^{\circ} $$

Find the exact value of the expression. $$ \sec \left(\sin ^{-1} \frac{12}{13}\right) $$

Find the area of the triangle whose sides have the given lengths. \(a=9, \quad b=12, \quad c=15\)

Find the exact value of the trigonometric function. $$ \sec \frac{17 \pi}{3} $$

Evaluate the expression without using a calculator. $$ \left(\cos 30^{\circ}\right)^{2}-\left(\sin 30^{\circ}\right)^{2} $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ \frac{3 \pi}{4} $$

Find the exact value of the expression. $$ \csc \left(\cos ^{-1} \frac{7}{25}\right) $$

Find the area of the triangle whose sides have the given lengths. \(a=1, \quad b=2, \quad c=2\)

Find the exact value of the trigonometric function. $$ \csc \frac{5 \pi}{4} $$

Evaluate the expression without using a calculator. $$ \left(\sin \frac{\pi}{3} \cos \frac{\pi}{4}-\sin \frac{\pi}{4} \cos \frac{\pi}{3}\right)^{2} $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ \frac{11 \pi}{6} $$

Find the exact value of the expression. $$ \tan \left(\sin ^{-1} \frac{12}{13}\right) $$

Find the area of the triangle whose sides have the given lengths. \(a=7, \quad b=8, \quad c=9\)

In triangle \(A B C, \angle A=40^{\circ}, a=15,\) and \(b=20\) (a) Show that there are two triangles, \(A B C\) and \(A^{\prime} B^{\prime} C,\) that satisfy these conditions. (b) Show that the areas of the triangles in part (a) are proportional to the sines of the angles \(C\) and \(C,\) that is, $$\frac{\text { area of } \triangle A B C}{\text { area of } \triangle A^{\prime} B^{\prime} C^{\prime}}=\frac{\sin C}{\sin C^{\prime}}$$

Find the exact value of the trigonometric function. $$ \cot \left(-\frac{\pi}{4}\right) $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ -\frac{\pi}{4} $$

Find the exact value of the expression. $$ \cot \left(\sin ^{-1} \frac{2}{3}\right) $$

Find the area of the triangle whose sides have the given lengths. \(a=11, \quad b=100, \quad c=101\)

Show that, given the three angles \(A, B, C\) of a triangle and one side, say \(a,\) the area of the triangle is $$ \text { area }=\frac{a^{2} \sin B \sin C}{2 \sin A} $$

Find the exact value of the trigonometric function. $$ \cos \frac{7 \pi}{4} $$

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ -45^{\circ} $$

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\sin ^{-1} x\right) $$

Tracking a Satellite The path of a satellite orbiting the earth causes it to pass directly over two tracking stations \(A\) and \(B,\) which are 50 \(\mathrm{mi}\) apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(87.0^{\circ}\) and \(84.2^{\circ}\) , respectively. (a) How far is the satellite from station \(A\) ? (b) How high is the satellite above the ground?

Find the exact value of the trigonometric function. $$ \tan \frac{5 \pi}{2} $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 70^{\circ}, \quad 430^{\circ} $$

Rewrite the expression as an algebraic expression in x. $$ \sin \left(\tan ^{-1} x\right) $$

Find the exact value of the trigonometric function. $$ \sin \frac{11 \pi}{6} $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ -30^{\circ}, \quad 330^{\circ} $$

Rewrite the expression as an algebraic expression in x. $$ \tan \left(\sin ^{-1} x\right) $$

Find the quadrant in which \(\theta\) lies from the information given. $$ \sin \theta<0 \quad \text { and } \quad \cos \theta<0 $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{5 \pi}{6}, \frac{17 \pi}{6} $$

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\tan ^{-1} x\right) $$

Distance Across a Lake Points \(A\) and \(B\) are separated by a lake. To find the distance between them, a surveyor locates a point \(C\) on land such that \(\angle C A B=48.6^{\circ} .\) He also measures \(C A\) as 312 \(\mathrm{ft}\) and \(C B\) as 527 ft. Find the distance between \(A\) and \(B .\)

Find the quadrant in which \(\theta\) lies from the information given. $$ \tan \theta<0 \quad \text { and } \quad \sin \theta<0 $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{32 \pi}{3}, \frac{11 \pi}{3} $$

Leaning Ladder \(A 20-f t\) ladder is leaning against a building. If the base of the ladder is 6 \(\mathrm{ft}\) from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?

The Leaning Tower of Pisa The bell tower of the cathedral in Pisa, Italy, leans \(5.6^{\circ}\) from the vertical. A tourist stands 105 \(\mathrm{m}\) from its base, with the tower leaning directly toward her. She measures the angle of elevation to the top of the tower to be \(29.2^{\circ} .\) Find the length of the tower to the nearest meter.

Three circles of radii 4, 5, and 6 cm are mutually tangent. Find the shaded area enclosed between the circles.

Find the quadrant in which \(\theta\) lies from the information given. $$ \sec \theta>0 \quad \text { and } \quad \tan \theta<0 $$

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ 155^{\circ}, \quad 875^{\circ} $$

Angle of the Sun A 96-ft tree casts a shadow that is 120 ft long. What is the angle of elevation of the sun?

Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 \(\mathrm{ft}\) and 180 \(\mathrm{ft}\) long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?