Question 1

Approximate the length using the formula for arc length. Round to the nearest meter.
-A television tower 540 m high subtends an angle of 3°50´. How far away is the tower?

Accepted Answer

The arc length formula is s = rθ, where θ is in radians. Convert 3°50' to radians, then use the formula with the height of the tower as the radius to find the distance.

Question 2

Solve the problem.
-A pendulum of length 12.1 inches swings 5°37´ to each side of its vertical position. What is the length (to the nearest hundredth of an inch)of the arc through which the end of the pendulum
Swings?

Accepted Answer

The answer of Solve the problem.
-A pendulum of length 12.1...

Question 3

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
-Find the distance between City E, 35° N and City F, 44° S. (Round to the nearest kilometer.)

Accepted Answer

The total distance between City E and City F can be calculated by considering the difference in latitude degrees and the circumference of the Earth. The Earth's circumference is approximately 2π * 6400 km. Since each degree of latitude is roughly equal to 111 km (the circumference of the Earth divided by 360), the distance between two points on the same meridian can be calculated as the difference in their latitudes times 111 km. City E is at 35° N and City F is at 44° S, making a total difference of 79° (35 + 44). Therefore, the distance is 79 * 111 km = 8769 km. However, since the question asks for the answer rounded to the nearest kilometer and none of the provided options exactly match this calculation, the closest option is A) 8824 km, indicating a slight approximation or rounding in the calculation process.

Question 4

Solve the problem.
-Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 45°, find the exact length of OU.

Accepted Answer

The answer of Solve the problem.
-Let angle POQ be designated...

Question 5

Find the exact value without using a calculator.
-$$\csc \frac { 5 \pi } { 3 }$$

Accepted Answer

The answer of Find the exact value without using a...

Question 6

Find the exact value of s in the given interval that has the given circular function value.
-$$\left[ \frac { 3 \pi } { 2 } , 2 \pi \right] ; \cos s = \frac { 1 } { 2 }$$

Accepted Answer

The answer of Find the exact value of s in...

Question 7

Find the exact value without using a calculator.
-$$\sin \frac { 3 \pi } { 4 }$$

Accepted Answer

The exact value of $\sin \frac { 3 \pi } { 4 }$ is $\frac { \sqrt { 2 } } { 2 }$, because $\frac { 3 \pi } { 4 }$ is in the second quadrant where sine is positive, and it corresponds to a 45-degree angle in standard position, whose sine is $\frac { \sqrt { 2 } } { 2 }$.

Question 8

Convert the degree measure to radians, correct to four decimal places. Use $$\text { Use } 3.1416 \text { for } \pi \text {. }$$
-31°

Accepted Answer

The answer of Convert the degree measure to radians, correct...

Question 9

Find the exact values of s in the given interval that satisfy the given condition.
-$$[ - \pi , \pi ) ; 2 \cos ^ { 2 } \mathrm {~s} = 1$$

Accepted Answer

The equation $2\cos^2 s = 1$ simplifies to $\cos^2 s = \frac{1}{2}$. Taking the square root of both sides gives $\cos s = \pm\frac{\sqrt{2}}{2}$, which corresponds to angles of $\pm\frac{\pi}{4}$ and $\pm\frac{3\pi}{4}$ within the given interval $[- \pi , \pi )$.

Question 10

Solve the problem.
-Find the radius of a circle in which a central angle of $\frac { \pi } { 3 }$ radian determines a sector of area 93 square meters. Round to the nearest hundredth.

Accepted Answer

The answer of Solve the problem.
-Find the radius of a...

Question 11

Convert the degree measure to radians. Leave answer as a multiple of $$\text { of } \pi \text {. }$$
--270°

Accepted Answer

To convert degrees to radians, use the conversion factor $\frac{\pi}{180}$. Thus, $-270° \times \frac{\pi}{180} = -\frac{270\pi}{180} = -\frac{3\pi}{2}$.

Question 12

Use the formula s = rωt to find the value of the missing variable. Give an exact answer.
-$$\mathrm { s } = \frac { \pi } { 3 } \mathrm {~m} , \mathrm { r } = 3 \mathrm {~m} , \mathrm { t } = 4 \mathrm { sec }$$

Accepted Answer

The formula $s = rωt$ can be rearranged to solve for $ω$ as $ω = \frac{s}{rt}$. Substituting the given values, $ω = \frac{\frac{\pi}{3}}{3 \times 4} = \frac{\pi}{3} \times \frac{1}{12} = \frac{\pi}{36}$ radians per sec.

Question 13

Convert the degree measure to radians, correct to four decimal places. Use $$\text { Use } 3.1416 \text { for } \pi \text {. }$$
-16.84°

Accepted Answer

The answer of Convert the degree measure to radians, correct...

Question 14

Solve the problem.
-A pulley with a diameter of 26 inches is driven by a belt which is moving 859 ft/min. To the nearest unit, how many revolutions per minute are made by the pulley?

Accepted Answer

The answer of Solve the problem.
-A pulley with a diameter...

Question 15

Find the exact circular function value.
-$\cos 2 \pi$

Accepted Answer

The cosine of $2\pi$ radians (which is equivalent to 360 degrees, a full circle) is 1. This is because $\cos 2\pi$ points to the same position on the unit circle as $\cos 0$, which is at (1, 0), thus having a cosine value of 1.

Question 16

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of .
-Find $\sin \theta$.

Accepted Answer

The answer of The figure shows an angle in standard...

Question 17

Solve the problem.
-A wheel with a 21-inch diameter is turning at the rate of 44 revolutions per minute. To the nearest inch, what is the speed of a point on the rim in in./min?

Accepted Answer

The answer of Solve the problem.
-A wheel with a 21-inch...

Question 18

Solve the problem.
-The angle of elevation $\theta$ of the sun in the sky at any latitude $L$ is calculated with the formula $\sin \theta = \cos \mathrm { D } \cos \mathrm { L } \cos \omega + \sin \mathrm { D } \sin \mathrm { L }$. $\omega$ is the number of radians that the Earth has rotated through since noon when $\omega = 0$. D is the declination of the sun. Westville has latitude $L = 36.5 ^ { \circ }$. Find the angle of elevation $\theta$ in degrees of the sun at 3 PM on February 29, 2012, where at that time $\mathrm { D } \approx - 0.1425$ and $\omega \approx 0.7854$.

Accepted Answer

To find the angle of elevation $\theta$, we substitute the given values into the formula $\sin \theta = \cos \mathrm { D } \cos \mathrm { L } \cos \omega + \sin \mathrm { D } \sin \mathrm { L }$. Given $L = 36.5 ^ { \circ }$, $D = -0.1425$, and $\omega = 0.7854$, we first convert $L$ to radians since trigonometric functions in calculators typically use radians. $36.5 ^ { \circ } = 0.637$ radians approximately. Substituting the values:$$\sin \theta = \cos(-0.1425) \cos(0.637) \cos(0.7854) + \sin(-0.1425) \sin(0.637)$$Calculating the right side of the equation gives us the sine of the angle $\theta$. To find $\theta$ itself, we take the arcsin (inverse sine) of the calculated value. Doing the math with the given values results in an angle of elevation $\theta$ close to $28.6 ^ { \circ }$, which corresponds to choice C.

Question 19

Convert the degree measure to radians, correct to four decimal places. Use $$\text { Use } 3.1416 \text { for } \pi \text {. }$$
-22°20´

Accepted Answer

The answer of Convert the degree measure to radians, correct...

Question 20

Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
-$$\frac { 3 \pi } { 8 }$$

Accepted Answer

The answer of Convert the radian measure to degrees. Round...

Quiz 3: Radian Measure and the Unit Circle

Approximate the length using the formula for arc length. Round to the nearest meter. -A television tower 540 m high subtends an angle of 3°50´. How far away is the tower?

Solve the problem. -A pendulum of length 12.1 inches swings 5°37´ to each side of its vertical position. What is the length (to the nearest hundredth of an inch) of the arc through which the end of the pendulum Swings?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the distance between City E, 35° N and City F, 44° S. (Round to the nearest kilometer.)

Find the exact value without using a calculator. - $\csc \frac { 5 \pi } { 3 }$

Find the exact value of s in the given interval that has the given circular function value. - $\left[ \frac { 3 \pi } { 2 } , 2 \pi \right] ; \cos s = \frac { 1 } { 2 }$

Find the exact value without using a calculator. - $\sin \frac { 3 \pi } { 4 }$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -31°

Find the exact values of s in the given interval that satisfy the given condition. - $[ - \pi , \pi ) ; 2 \cos ^ { 2 } \mathrm {~s} = 1$

Solve the problem. -Find the radius of a circle in which a central angle of $\frac { \pi } { 3 }$ radian determines a sector of area 93 square meters. Round to the nearest hundredth.

Convert the degree measure to radians. Leave answer as a multiple of $\text { of } \pi \text {. }$ --270°

Use the formula s = rωt to find the value of the missing variable. Give an exact answer. - $\mathrm { s } = \frac { \pi } { 3 } \mathrm {~m} , \mathrm { r } = 3 \mathrm {~m} , \mathrm { t } = 4 \mathrm { sec }$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -16.84°

Solve the problem. -A pulley with a diameter of 26 inches is driven by a belt which is moving 859 ft/min. To the nearest unit, how many revolutions per minute are made by the pulley?

Find the exact circular function value. - $\cos 2 \pi$

Solve the problem. -A wheel with a 21-inch diameter is turning at the rate of 44 revolutions per minute. To the nearest inch, what is the speed of a point on the rim in in./min?

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -22°20´

Convert the radian measure to degrees. Round to the nearest hundredth if necessary. - $\frac { 3 \pi } { 8 }$

Quiz 3: Radian Measure and the Unit Circle

Approximate the length using the formula for arc length. Round to the nearest meter. -A television tower 540 m high subtends an angle of 3°50´. How far away is the tower?

Solve the problem. -A pendulum of length 12.1 inches swings 5°37´ to each side of its vertical position. What is the length (to the nearest hundredth of an inch) of the arc through which the end of the pendulum Swings?

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the distance between City E, 35° N and City F, 44° S. (Round to the nearest kilometer.)

Solve the problem. -Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 45°, find the exact length of OU.

Find the exact value without using a calculator. - csc⁡5π3\csc \frac { 5 \pi } { 3 }csc35π​

Find the exact value of s in the given interval that has the given circular function value. - [3π2,2π];cos⁡s=12\left[ \frac { 3 \pi } { 2 } , 2 \pi \right] ; \cos s = \frac { 1 } { 2 }[23π​,2π];coss=21​

Find the exact value without using a calculator. - sin⁡3π4\sin \frac { 3 \pi } { 4 }sin43π​

Convert the degree measure to radians, correct to four decimal places. Use Use 3.1416 for π. \text { Use } 3.1416 \text { for } \pi \text {. } Use 3.1416 for π. -31°

Find the exact values of s in the given interval that satisfy the given condition. - [−π,π) ;2cos⁡2 s=1[ - \pi , \pi ) ; 2 \cos ^ { 2 } \mathrm {~s} = 1[−π,π) ;2cos2 s=1

Solve the problem. -Find the radius of a circle in which a central angle of π3\frac { \pi } { 3 }3π​ radian determines a sector of area 93 square meters. Round to the nearest hundredth.

Convert the degree measure to radians. Leave answer as a multiple of of π. \text { of } \pi \text {. } of π. --270°

Use the formula s = rωt to find the value of the missing variable. Give an exact answer. - s=π3 m,r=3 m,t=4sec\mathrm { s } = \frac { \pi } { 3 } \mathrm {~m} , \mathrm { r } = 3 \mathrm {~m} , \mathrm { t } = 4 \mathrm { sec }s=3π​ m,r=3 m,t=4sec

Convert the degree measure to radians, correct to four decimal places. Use Use 3.1416 for π. \text { Use } 3.1416 \text { for } \pi \text {. } Use 3.1416 for π. -16.84°

Solve the problem. -A pulley with a diameter of 26 inches is driven by a belt which is moving 859 ft/min. To the nearest unit, how many revolutions per minute are made by the pulley?

Find the exact circular function value. - cos⁡2π\cos 2 \picos2π

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated circular function value of . -Find sin⁡θ\sin \thetasinθ .

Solve the problem. -A wheel with a 21-inch diameter is turning at the rate of 44 revolutions per minute. To the nearest inch, what is the speed of a point on the rim in in./min?

Convert the degree measure to radians, correct to four decimal places. Use Use 3.1416 for π. \text { Use } 3.1416 \text { for } \pi \text {. } Use 3.1416 for π. -22°20´

Convert the radian measure to degrees. Round to the nearest hundredth if necessary. - 3π8\frac { 3 \pi } { 8 }83π​

Find the exact value without using a calculator. - $\csc \frac { 5 \pi } { 3 }$

Find the exact value of s in the given interval that has the given circular function value. - $\left[ \frac { 3 \pi } { 2 } , 2 \pi \right] ; \cos s = \frac { 1 } { 2 }$

Find the exact value without using a calculator. - $\sin \frac { 3 \pi } { 4 }$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -31°

Find the exact values of s in the given interval that satisfy the given condition. - $[ - \pi , \pi ) ; 2 \cos ^ { 2 } \mathrm {~s} = 1$

Solve the problem. -Find the radius of a circle in which a central angle of $\frac { \pi } { 3 }$ radian determines a sector of area 93 square meters. Round to the nearest hundredth.

Convert the degree measure to radians. Leave answer as a multiple of $\text { of } \pi \text {. }$ --270°

Use the formula s = rωt to find the value of the missing variable. Give an exact answer. - $\mathrm { s } = \frac { \pi } { 3 } \mathrm {~m} , \mathrm { r } = 3 \mathrm {~m} , \mathrm { t } = 4 \mathrm { sec }$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -16.84°

Find the exact circular function value. - $\cos 2 \pi$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -22°20´

Convert the radian measure to degrees. Round to the nearest hundredth if necessary. - $\frac { 3 \pi } { 8 }$