Chapter 9

Answers are given at the end of these exercises. Write the formula for the distance \(d\) from \(P_{1}=\left(x_{1}, y_{1}\right)\) to \(P_{2}=\left(x_{2}, y_{2}\right)\)

The amplitude \(A\) and period \(T\) of \(f(x)=5 \sin (4 x)\) are ______ and _______.

The area \(K\) of a triangle whose base is \(b\) and whose altitude is \(h\) is ________

In a right triangle, if the length of the hypotenuse is 65 and the length of one of the other sides is \(63,\) what is the length of the third side?

If \(\theta\) is an acute angle, solve the equation \(\cos \theta=\frac{\sqrt{2}}{2}\).

Approximate the angular speed of the second hand on a clock in rad/sec. (Round to three decimal places.)

True or False $$\cos ^{2} \frac{\theta}{2}=\frac{1+\sin \theta}{2}$$

Solve \(\sin A=\frac{1}{2}\) if \(0 \leq A \leq \pi\).

Write an equation for a sine function with period 12 and amplitude 7.

Find the length of the arc of a circle of radius 5 feet subtended by a central angle of 2.7 radians.

Multiple Choice If one side and two angles of a triangle are known, which law can be used to solve the triangle? (a) Law of Sines (b) Law of Cosines (c) Either a or b (d) The triangle cannot be solved.

If \(\theta\) is an acute angle, solve the equation \(\tan \theta=\frac{1}{2} .\) Express your answer in degrees, rounded to one decimal place.

Multiple Choice If two sides and the included angle of a triangle are known, which law can be used to solve the triangle? (a) Law of Sines (b) Law of Cosines (c) Either a or b (d) The triangle cannot be solved.

Find the area of the right triangle whose legs are of length 3 and 4

If none of the angles of a triangle is a right angle, the triangle is called _________. (a) oblique (b) obtuse (c) acute (d) scalene

Find the exact values of \(\sin ^{-1} \frac{1}{2}\) and \(\tan ^{-1} 1 .\) Express your answer in degrees.

True or False If the distance \(d\) of an object from its rest position at time \(t\) is given by a sinusoidal graph, the motion of the object is simple harmonic motion.

True or False The area of a triangle equals one-half the product of the lengths of two of its sides times the sine of their included angle

In Problems 7-10, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=5 ; \quad T=2 \text { seconds } $$

If two angles of a triangle measure \(48^{\circ}\) and \(93^{\circ},\) what is the measure of the third angle? (a) \(132^{\circ}\) (b) \(77^{\circ}\) (c) \(42^{\circ}\) (d) \(39^{\circ}\)

The sum of the measures of the two acute angles in a right triangle is _____. (a) \(45^{\circ}\) (b) \(90^{\circ}\) (c) \(180^{\circ}\) (d) \(360^{\circ}\)

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=10 ; \quad T=3 \text { seconds } $$

True or False A special case of the Law of Cosines is the Pythagorean Theorem.

Heron's Formula is used to find the area of (a) ASA (b) SAS (c) SSS (d) AAS

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=7 ; \quad T=5 \pi \text { seconds } $$

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, find a function that relates the displacement d of the object from its rest position after t seconds. Assume that the positive direction of the motion is up. $$ a=4 ; \quad T=\frac{\pi}{2} \text { seconds } $$

In Problems \(15-22,\) the displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=5 \sin (3 t) $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=4 \sin (2 t) $$

In Problems 17-32, solve each triangle. $$ a=3, \quad b=4, \quad C=40^{\circ} $$

Find the area of each triangle. Round answers to two decimal places. $$a=3, \quad b=4, \quad C=50^{\circ}$$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=5 \cos \left(\frac{\pi}{2} t\right) $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=-9 \sin \left(\frac{1}{4} t\right) $$

Solve each triangle. $$ A=55^{\circ}, \quad B=25^{\circ}, \quad a=4 $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=-2 \cos (2 t) $$

Solve each triangle. $$ a=6, \quad b=4, \quad C=60^{\circ} $$

Find the area of each triangle. Round answers to two decimal places. $$a=6, \quad b=4, \quad C=60^{\circ}$$

Solve each triangle. $$ A=50^{\circ}, \quad C=20^{\circ}, \quad a=3 $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=3+7 \cos (3 \pi t) $$

The displacement \(d\) (in meters) of an object at time \(t\) (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is the time required for one oscillation? (d) What is the frequency? $$ d(t)=4+3 \sin (\pi t) $$

Solve each triangle. $$ b=4, \quad c=1, \quad A=120^{\circ} $$

Find the area of each triangle. Round answers to two decimal places. $$b=4, \quad c=1, \quad A=120^{\circ}$$

In Problems 23-26, graph each damped vibration curve for \(0 \leq t \leq 2 \pi\). $$ d(t)=e^{-t / \pi} \cos (2 t) $$

Find the area of each triangle. Round answers to two decimal places. $$a=12, \quad b=35, \quad c=37$$

Solve each triangle. $$ A=110^{\circ}, \quad C=30^{\circ}, \quad c=3 $$

Graph each damped vibration curve for \(0 \leq t \leq 2 \pi\). $$ d(t)=e^{-t / 2 \pi} \cos (2 t) $$

Solve each triangle. $$ a=3, \quad c=2, \quad B=90^{\circ} $$

Find the area of each triangle. Round answers to two decimal places. $$a=4, \quad b=4, \quad c=4$$

The hypotenuse of a right triangle is 5 inches. If one leg is 2 inches, find the degree measure of each angle.

Graph each damped vibration curve for \(0 \leq t \leq 2 \pi\). $$ d(t)=e^{-t / 4 \pi} \cos t $$

Solve each triangle. $$ a=4, \quad b=5, \quad c=3 $$