5.R: Trigonometric Functions (Review)

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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: $alt$

12) $75°$

13) $\dfrac{5π}{4}$

Answer: $CNX_Precalc_Figure_05_04_219.jpg$

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.

$alt$

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)

$alt$

Answer: $a=4, b=4 $

11)

$alt$

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: $alt$

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.

$alt$

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.