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Mathematics LibreTexts

5.R: Trigonometric Functions (Review)

  • Page ID
    18195
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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

    CNX_Precalc_Figure_05_04_217.jpg

    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.

    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.

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