5.R: Trigonometric Functions (Review)

5.1: Review Exercises

For the exercises 1-2, convert the angle measures to degrees.

1) $\dfrac{π}{4}$

Answer

$45°$

2) $−\dfrac{5π}{3}$

For the exercises 3-6, convert the angle measures to radians.

3) $-210°$

Answer

$−\dfrac{7π}{6}$

4) $180°$

5) Find the length of an arc in a circle of radius $7$ meters subtended by the central angle of $85°$.

Answer

$10.385$ meters

6) Find the area of the sector of a circle with diameter $32$ feet and an angle of $\dfrac{3π}{5}$ radians.

For the exercises 7-8, find the angle between $0°$ and $360°$ that is coterminal with the given angle.

7) $420°$

Answer

$60°$

8) $−80°$

For the exercises 9-10, find the angle between $0$ and $2π$ in radians that is coterminal with the given angle.

9) $− \dfrac{20π}{11}$

Answer

$\dfrac{2π}{11}$

10) $\dfrac{14π}{5}$

For the exercises 11-, draw the angle provided in standard position on the Cartesian plane.

11) $-210°$

Answer

12) $75°$

13) $\dfrac{5π}{4}$

Answer

14) $−\dfrac{π}{3}$

15) Find the linear speed of a point on the equator of the earth if the earth has a radius of $3,960$ miles and the earth rotates on its axis every $24$ hours. Express answer in miles per hour.

Answer

$1036.73$ miles per hour

16) A car wheel with a diameter of $18$ inches spins at the rate of $10$ revolutions per second. What is the car's speed in miles per hour?

5.2: Review Exercises

1) Find the exact value of $\sin \dfrac{π}{3}$.

Answer

$\dfrac{\sqrt{3}}{2}$

2) Find the exact value of $\cos \dfrac{π}{4}$.

3) Find the exact value of $\cos π$.

Answer

$-1$

4) State the reference angle for $300°$.

5) State the reference angle for $\dfrac{3π}{4}$.

Answer

$\dfrac{π}{4}$

6) Compute cosine of $330°$.

7) Compute sine of $\dfrac{5π}{4}$.

Answer

$−\dfrac{\sqrt{2}}{2}$

8) State the domain of the sine and cosine functions.

9) State the range of the sine and cosine functions.

Answer

$[–1,1]$

5.3: Review Exercises

For the exercises 1-4, find the exact value of the given expression.

1) $\cos \dfrac{π}{6}$

2) $\tan \dfrac{π}{4}$

Answer

$1$

3) $\csc \dfrac{π}{3}$

4) $\sec \dfrac{π}{4}$

Answer

$\sqrt{2}$

For the exercises 4-12, use reference angles to evaluate the given expression.

5) $\sec \dfrac{11π}{3}$

6) $\sec 315°$

Answer

$\sqrt{2}$

7) If $\sec (t)=−2.5$, what is the $\sec (−t)$?

8) If $\tan (t)=−0.6$, what is the $\tan (−t)$?

Answer

$0.6$

9) If $\tan (t)=\dfrac{1}{3}$, find $\tan (t−π)$.

10) If $\cos (t)= \dfrac{\sqrt{2}}{2}$, find $\sin (t+2π)$.

Answer

$\dfrac{\sqrt{2}}{2}$ or $−\dfrac{\sqrt{2}}{2}$

11) Which trigonometric functions are even?

12) Which trigonometric functions are odd?

Answer

sine, cosecant, tangent, cotangent

5.4: Review Exercises

For the exercises 1-5, use side lengths to evaluate.

1) $\cos \dfrac{π}{4}$

2) $\cot \dfrac{π}{3}$

Answer

$\dfrac{\sqrt{3}}{3}$

3) $\tan \dfrac{π}{6}$

4) $\cos (\dfrac{π}{2}) = \sin ( \_\_°)$

Answer

$0$

5) $\csc (18°)= \sec (\_\_°)$

For the exercises 6-7, use the given information to find the lengths of the other two sides of the right triangle.

6) $\cos B= \dfrac{3}{5}, a=6$

Answer

$b=8,c=10$

7) $\tan A = \dfrac{5}{9},b=6$

For the exercises 8-9, use Figure below to evaluate each trigonometric function.

8) $\sin A$

Answer

$\dfrac{11\sqrt{157}}{157}$

9) $\tan B$

For the exercises 10-11, solve for the unknown sides of the given triangle.

10)

Answer

$a=4, b=4$

11)

12) A $15$-ft ladder leans against a building so that the angle between the ground and the ladder is $70°$. How high does the ladder reach up the side of the building?

Answer

$14.0954$ ft

13) The angle of elevation to the top of a building in Baltimore is found to be $4$ degrees from the ground at a distance of $1$ mile from the base of the building. Using this information, find the height of the building.

Practice Test

1) Convert $\dfrac{5π}{6}$ radians to degrees.

Answer

$150°$

2) Convert $−620°$ to radians.

3) Find the length of a circular arc with a radius $12$ centimeters subtended by the central angle of $30°$.

Answer

$6.283$ centimeters

4) Find the area of the sector with radius of $8$ feet and an angle of $\dfrac{5π}{4}$ radians.

5) Find the angle between $0°$ and $360°$ that is coterminal with $375°$.

Answer

$15°$

6) Find the angle between $0$ and $2π$ in radians that is coterminal with $−\dfrac{4π}{7}$.

7) Draw the angle $315°$ in standard position on the Cartesian plane.

Answer

8) Draw the angle $−\dfrac{π}{6}$ in standard position on the Cartesian plane.

9) A carnival has a Ferris wheel with a diameter of $80$ feet. The time for the Ferris wheel to make one revolution is $75$ seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

Answer

$3.351$ feet per second, $\dfrac{2π}{75}$ radians per second

10) Find the exact value of $\sin \dfrac{π}{6}$.

11) Compute sine of $240°$.

Answer

$−\dfrac{\sqrt{3}}{2}$

12) State the domain of the sine and cosine functions.

13) State the range of the sine and cosine functions.

Answer

$[ –1,1 ]$

14) Find the exact value of $\cot \dfrac{π}{4}$.

15) Find the exact value of $\tan \dfrac{π}{3}$.

Answer

$\sqrt{3}$

16) Use reference angles to evaluate $\csc \dfrac{7π}{4}$.

17) Use reference angles to evaluate $\tan 210°$.

Answer

$\dfrac{\sqrt{3}}{3}$

18) If $\csc t=0.68$, what is the $\csc (−t)$?

19) If $\cos t= \dfrac{\sqrt{3}}{2}$, find $\cos (t−2π)$.

Answer

$\dfrac{\sqrt{3}}{2}$

20) Which trigonometric functions are even?

21) Find the missing angle: $\cos \left(\dfrac{\pi }{6} \right)= \sin (\;)$

Answer

$\dfrac{π}{3}$

22) Find the missing sides of the triangle $ABC: \sin B= \dfrac{3}{4},c=12$

23) Find the missing sides of the triangle.

Answer

$a=\dfrac{9}{2},b=\dfrac{9\sqrt{3}}{2}$

24) The angle of elevation to the top of a building in Chicago is found to be $9$ degrees from the ground at a distance of $2000$ feet from the base of the building. Using this information, find the height of the building.