Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

5.E: Trigonometric Functions (Exercises)

[ "article:topic", "authorname:openstax", "license:ccby", "showtoc:no" ]
  • Page ID
    5972
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    5.1: Angles

    Section Exercises

    Verbal

    Draw an angle in standard position. Label the vertex, initial side, and terminal side.

    Explain why there are an infinite number of angles that are coterminal to a certain angle.

    State what a positive or negative angle signifies, and explain how to draw each.

    Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

    How does radian measure of an angle compare to the degree measure? Include an explanation of 1 radian in your paragraph.

    Explain the differences between linear speed and angular speed when describing motion along a circular path.

    Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

    Graphical

    For the following exercises, draw an angle in standard position with the given measure.

    30°

    300°

     

    −80°

    135°

    −150°

    \(\frac{2π}{3}\)

    \(\frac{7π}{4}\)

    \(\frac{5π}{6}\)

    \(\frac{π}{2}\)

    \(−\frac{π}{10}\)

    415°

    −120°

    240°

    −315°

    \(\frac{22π}{3}\)

    \(\frac{4π}{3}\)

     

    \(−\frac{π}{6}\)

    \(−\frac{4π}{3}\)

    \(\frac{2π}{3}\)

    For the following exercises, refer to Figure. Round to two decimal places.

     

    Find the arc length.

    Find the area of the sector.

    \(\frac{27π}{2}≈11.00 \text{ in}^2\)

    For the following exercises, refer to Figure. Round to two decimal places.

    Find the arc length.

    Find the area of the sector.

    \(\frac{81π}{20}≈12.72\text{ cm}^2\)

    Algebraic

    For the following exercises, convert angles in radians to degrees.

    \(\frac{3π}{4}\) radians

    \(\frac{π}{9}\) radians

    20°

    \(−\frac{5π}{4}\) radians

    \(\frac{π}{3}\) radians

    60°

    \(−\frac{7π}{3}\) radians

    \(−\frac{5π}{12}\) radians

    −75°

    \(\frac{11π}{6}\) radians

    For the following exercises, convert angles in degrees to radians.

    90°

    \(\frac{π}{2}\) radians

    100°

    −540°

    \(−3π\) radians

    −120°

    180°

    \(π\) radians

    −315°

    150°

    \(\frac{5π}{6}\) radians

    For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places.

    Find the length of the arc of a circle of radius 12 inches subtended by a central angle of \(\frac{π}{4}\) radians.

    Find the length of the arc of a circle of radius 5.02 miles subtended by the central angle of \(\frac{π}{3}\).

    \(\frac{5.02π}{3}≈5.26\) miles

    Find the length of the arc of a circle of diameter 14 meters subtended by the central angle of 5π6.5π6.

    Find the length of the arc of a circle of radius 10 centimeters subtended by the central angle of 50°.

    \(\frac{25π}{9}≈8.73\) centimeters

    Find the length of the arc of a circle of radius 5 inches subtended by the central angle of 220°.

    Find the length of the arc of a circle of diameter 12 meters subtended by the central angle is 63°.

    \(\frac{21π}{10}≈6.60\) meters

    For the following exercises, use the given information to find the area of the sector. Round to four decimal places.

    A sector of a circle has a central angle of 45° and a radius 6 cm.

    A sector of a circle has a central angle of 30° and a radius of 20 cm.

    104.7198 cm2

    A sector of a circle with diameter 10 feet and an angle of \(\frac{π}{2}\) radians.

    A sector of a circle with radius of 0.7 inches and an angle of \(π\) radians.

    0.7697 in

    or the following exercises, find the angle between 0° and 360° that is coterminal to the given angle.

    −40°

    −110°

    250°

    700°

    1400°

    320°

    For the following exercises, find the angle between 0 and 2π 2π in radians that is coterminal to the given angle.

    \(−\frac{π}{9}\)

    \(\frac{10π}{3}\)

    \(\frac{4π}{3}\)

    \(\frac{13π}{6}\)

    \(\frac{44π}{9}\)

    \(\frac{8π}{9}\)

    Real-World Applications

    A truck with 32-inch diameter wheels is traveling at 60 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

    A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

    1320 rad 210.085 RPM

    A wheel of radius 8 inches is rotating 15°/s. What is the linear speed \(v\), the angular speed in RPM, and the angular speed in rad/s?

    A wheel of radius 14 inches is rotating 0.5 rad/s. What is the linear speed \(v\), the angular speed in RPM, and the angular speed in deg/s?

    7 in./s, 4.77 RPM, 28.65 deg/s

    A CD has diameter of 120 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 200 RPM (revolutions per minute). Find the linear speed.

    When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters.

    \(1,809,557.37 \text{ mm/min}=30.16 \text{ m/s}\)

    A person is standing on the equator of Earth (radius 3960 miles). What are his linear and angular speeds?

    Find the distance along an arc on the surface of Earth that subtends a central angle of 5 minutes \((1 \text{ minute}=\frac{1}{60} \text{ degree})\). The radius of Earth is 3960 miles.

    \(5.76\) miles

    Find the distance along an arc on the surface of Earth that subtends a central angle of 7 minutes \((1 \text{ minute}=\frac{1}{60} \text{ degree})\). The radius of Earth is \(3960\) miles.

    Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in \(20\) minutes?

    \(120°\)

    Extensions

    Two cities have the same longitude. The latitude of city A is 9.00 degrees north and the latitude of city B is 30.00 degree north. Assume the radius of the earth is 3960 miles. Find the distance between the two cities.

    A city is located at 40 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

    794 miles per hour

    A city is located at 75 degrees north latitude. Assume the radius of the earth is 3960 miles and the earth rotates once every 24 hours. Find the linear speed of a person who resides in this city.

    Find the linear speed of the moon if the average distance between the earth and moon is 239,000 miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

    2,234 miles per hour

    A bicycle has wheels 28 inches in diameter. A tachometer determines that the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is travelling down the road.

    A car travels 3 miles. Its tires make 2640 revolutions. What is the radius of a tire in inches?

    11.5 inches

    A wheel on a tractor has a 24-inch diameter. How many revolutions does the wheel make if the tractor travels 4 miles?

    Review Exercises

    For the following exercises, convert the angle measures to degrees.

    \(\frac{π}{4}\) 

    \(45°\)

    \(−\frac{5π}{3}\)

    For the following exercises, convert the angle measures to radians.

    -210°

    \(−\frac{7π}{6}\)

    180°

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

    10.385 meters

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

    For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.

    \(420°\)

    \(60°\)

    \(−80°\)

    For the following exercises, find the angle between 0 and \(2π\) in radians that is coterminal with the given angle.

    \(− \frac{20π}{11}\)

    \(\frac{2π}{11}\)

    \(\frac{14π}{5}\)

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

    -210°

    75°

    \(\frac{5π}{4}\)

    \(−\frac{π}{3}\)

    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.

    1036.73 miles per hour

    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: Unit Circle - Sine and Cosine Functions

    Section Exercises

    Verbal

    Describe the unit circle.

    The unit circle is a circle of radius 1 centered at the origin.

    What do the x- and y-coordinates of the points on the unit circle represent?

    Discuss the difference between a coterminal angle and a reference angle.

    Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, \(t\), formed by the terminal side of the angle \(t\) and the horizontal axis.

    Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

    Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

    The sine values are equal.

     

    Algebraic

    For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by ttlies.

    \( \sin (t)<0\) and \( \cos (t)<0\)

    \( \sin (t)>0\) and \( \cos (t)>0\)

    I

    \( \sin (t)>0 \) and  \( \cos (t)<0\)

    \( \sin (t)<0 \) and \( \cos (t)>0\)

    IV

    For the following exercises, find the exact value of each trigonometric function.

    \(\sin \frac{π}{2}\)

    \(\sin \frac{π}{3}\)

    \(\frac{\sqrt{3}}{2}\)

    \( \cos \frac{π}{2}\)

    \( \cos \frac{π}{3}\)

    \(\frac{1}{2}\)

    \( \sin \frac{π}{4}\)

    \( \cos \frac{π}{4}\)

    \(\frac{\sqrt{2}}{2}\)

    \( \sin \frac{π}{6}\)

    \( \sin π\)

    0

    \( \sin \frac{3π}{2}\)

    \( \cos π\)

    −1

    \( \cos 0\)

    \(cos \frac{π}{6}\)

    \(\frac{\sqrt{3}}{2}\)

    \( \sin 0\)

     

    Numeric

    For the following exercises, state the reference angle for the given angle.

    \(240°\)

    \(60°\)

    \(−170°\)

    \(100°\)

    \(80°\)

    \(−315°\)

    \(135°\)

    \(45°\)

    \(\frac{5π}{4}\)

    \(\frac{2π}{3}\)

    \(\frac{π}{3}\)

    \(\frac{5π}{6}\)

    \(−\frac{11π}{3}\)

    \(\frac{π}{3}\)

    \(\frac{−7π}{4}\)

    \(\frac{−π}{8}\)

    \(\frac{π}{8}\)

     

    For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

    \(225°\)

    \(300°\)

    \(60°\), Quadrant IV, \( \sin (300°)=−\frac{\sqrt{3}}{2}, \cos (300°)=\frac{1}{2}\)

    \(320°\)

    \(135°\)

    \(45°\), Quadrant II,  \( \sin (135°)=\frac{\sqrt{2}}{2}, \cos (135°)=−\frac{\sqrt{2}}{2}\)

    \(210°\)

    \(120°\)

    \(60°\), Quadrant II,  \(\sin (120°)=\frac{\sqrt{3}}{2}\), \(\cos (120°)=−\frac{1}{2}\)

    \(250°\)

    \(150°\)

    \(30°\), Quadrant II,  \( \sin (150°)=\frac{1}{2}\), \cos(150°)=−\frac{\sqrt{3}}{2}\)

    \(\frac{5π}{4}\)

    \(\frac{7π}{6}\)

    \(\frac{π}{6}\), Quadrant III,  \(\sin( \frac{7π}{6})=−\frac{1}{2}\), \(\cos(\frac{7π}{6})=−\frac{\sqrt{3}}{2}\)

    \(\frac{5π}{3}\)

    \(\frac{3π}{4}\)

    \(\frac{π}{4}\), Quadrant II,  \(\sin (\frac{3π}{4})=\frac{\sqrt{2}}{2}\), \(\cos(\frac{4π}{3})=−\frac{\sqrt{2}}{2}\)

    \(\frac{4π}{3}\)

    \(\frac{2π}{3}\)

    \(\frac{π}{3}\), Quadrant II,  \( \sin (\frac{2π}{3})=\frac{\sqrt{3}}{2}\), \( \cos (\frac{2π}{3})=−\frac{1}{2}\)

    \(\frac{5π}{6}\)

    \(\frac{7π}{4}\)

    \(\frac{π}{4}\), Quadrant IV,  \( \sin (\frac{7π}{4})=−\frac{\sqrt{2}}{2}\), \( \cos (\frac{7π}{4})=\frac{\sqrt{2}}{2}\)

     

    For the following exercises, find the requested value.

    If \(\cos (t)=\frac{1}{7}\) and \(t\) is in the 4th quadrant, find \( \sin (t)\).

    If \( \cos (t)=\frac{2}{9}\) and \(t\) is in the 1st quadrant, find \(\sin (t).\)

    \(\frac{\sqrt{77}}{9}\)

    If \(\sin (t)=\frac{3}{8}\) and \(t\) is in the 2nd quadrant, find \( \cos (t)\).

    If \( \sin (t)=−\frac{1}{4}\) and \(t\) is in the 3rd quadrant, find \(\cos (t).\)

    \(−\frac{\sqrt{15}}{4}\)

    Find the coordinates of the point on a circle with radius 15 corresponding to an angle of \(220°\).

    Find the coordinates of the point on a circle with radius 20 corresponding to an angle of \(120°\).

    \((−10,10\sqrt{3})\)

    Find the coordinates of the point on a circle with radius 8 corresponding to an angle of \(\frac{7π}{4}\).

    Find the coordinates of the point on a circle with radius 16 corresponding to an angle of \(\frac{5π}{9}\).

    \((–2.778,15.757)\)

    State the domain of the sine and cosine functions.

    State the range of the sine and cosine functions.

    \([–1,1]\)

     

    Graphical

    For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of \(t\) .

     

    \( \sin t=\frac{1}{2}, \cos t=−\frac{\sqrt{3}}{2}\)

     

    \( \sin t=− \frac{\sqrt{2}}{2}, \cos t=−\frac{\sqrt{2}}{2}\)

     

    \( \sin t=\frac{\sqrt{3}}{2},\cos t=−\frac{1}{2}\)

     

    \( \sin t=− \frac{\sqrt{2}}{2}, \cos t=\frac{\sqrt{2}}{2}\)

     

    \( \sin t=0, \cos t=−1\)

     

    \( \sin t=−0.596, \cos t=0.803\)

     

    \( \sin t=\frac{1}{2}, \cos t= \frac{\sqrt{3}{2}}\)

     

    \( \sin t=−\frac{1}{2}, \cos t= \frac{\sqrt{3}}{2} \)

     

    \( \sin t=0.761, \cos t=−0.649 \)

     

    \( \sin t=1, \cos t=0\)

    Technology

    For the following exercises, use a graphing calculator to evaluate.

    \( \sin \frac{5π}{9}\)

    \(cos \frac{5π}{9}\)

    −0.1736

    \( \sin \frac{π}{10}\)

    \( \cos \frac{π}{10}\)

    0.9511

    \( \sin \frac{3π}{4}\)

    \(\cos \frac{3π}{4}\)

    −0.7071

    \( \sin 98° \)

    \( \cos 98° \)

    −0.1392

    \( \cos 310° \)

    \( \sin 310° \)

    −0.7660

    Extensions

    For the following exercises, evaluate.

    \( \sin (\frac{11π}{3}) \cos (\frac{−5π}{6})\)

    \( \sin (\frac{3π}{4}) \cos (\frac{5π}{3}) \)

    \(\frac{\sqrt{2}}{4}\)

    \( \sin (− \frac{4π}{3}) \cos (\frac{π}{2})\)

    \( \sin (\frac{−9π}{4}) \cos (\frac{−π}{6})\)

    \(−\frac{\sqrt{6}}{4}\)

    \( \sin (\frac{π}{6}) \cos (\frac{−π}{3}) \)

    \( \sin (\frac{7π}{4}) \cos (\frac{−2π}{3}) \)

    \(\frac{\sqrt{2}}{4}\)

    \( \cos (\frac{5π}{6}) \cos (\frac{2π}{3})\)

    \( \cos (\frac{−π}{3}) \cos (\frac{π}{4}) \)

    \(\frac{\sqrt{2}}{4}\)

    \( \sin (\frac{−5π}{4}) \sin (\frac{11π}{6})\)

    \( \sin (π) \sin (\frac{π}{6}) \)

    0

    Real-World Applications

    For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point \((0,1)\), that is, on the due north position. Assume the carousel revolves counter clockwise.

    What are the coordinates of the child after 45 seconds?

    What are the coordinates of the child after 90 seconds?

    \((0,–1)\)

    What is the coordinates of the child after 125 seconds?

    When will the child have coordinates \((0.707,–0.707)\)  if the ride lasts 6 minutes? (There are multiple answers.)

    37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

    When will the child have coordinates \((−0.866,−0.5)\)  if the ride last 6 minutes?

    Review Exercises

    Find the exact value of \( \sin \frac{π}{3}\).

    \(\frac{\sqrt{3}}{2}\)

    Find the exact value of \( \cos \frac{π}{4}\).

    Find the exact value of \( \cos π \).

    –1

    State the reference angle for \(300°\).

    State the reference angle for \( \frac{3π}{4}\).

    \( \frac{π}{4}\)

    Compute cosine of \(330°\).

    Compute sine of \(\frac{5π}{4}\).

    \(−\frac{\sqrt{2}}{2}\)

    State the domain of the sine and cosine functions.

    State the range of the sine and cosine functions.

    \([–1,1]\)

    5.3: The Other Trigonometric Functions

    Section Exercises

    Verbal

    On an interval of \([ 0,2π )\), can the sine and cosine values of a radian measure ever be equal? If so, where?

    Yes, when the reference angle is \(\frac{π}{4}\) and the terminal side of the angle is in quadrants I and III. Thus, at \(x=\frac{π}{4},\frac{5π}{4}\), the sine and cosine values are equal.

    What would you estimate the cosine of π π degrees to be? Explain your reasoning.

    For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

    Substitute the sine of the angle in for \(y\) in the Pythagorean Theorem \(x^2+y^2=1\). Solve for \(x\) and take the negative solution.

    Describe the secant function.

    Tangent and cotangent have a period of \(π\).  What does this tell us about the output of these functions?

    The outputs of tangent and cotangent will repeat every \(π\)  units.

    Algebraic

    For the following exercises, find the exact value of each expression.

    \( \tan \; \frac{π}{6}\)

    \(\sec \; \frac{π}{6}\)

    \(\frac{2\sqrt{3}}{3}\)

    \( \csc \; \csc \; \frac{π}{6}\)

    \( \cot \; \frac{π}{6}\)

    \(\sqrt{3}\)

    \( \tan \; \frac{π}{4}\)

    \( \sec \; \frac{π}{4}\)

    \(\sqrt{2}\)

    \( \csc \; \csc \; \frac{π}{4}\)

    \( \cot \; \frac{π}{4}\)

    1

    \( \tan \; \frac{π}{3}\)

    \( \sec \; \frac{π}{3}\)

    2

    \( \csc \; \csc \; \frac{π}{3}\)

    \( \cot \; \frac{π}{3}\)

    \(\frac{\sqrt{3}}{3}\)

    For the following exercises, use reference angles to evaluate the expression.

    \( \tan \; \frac{5π}{6}\)

    \( \sec \; \frac{7π}{6}\)

    \(−\frac{2\sqrt{3}}{3}\)

    \( \csc \; \csc \; \frac{11π}{6}\)

    \( \cot \; \frac{13π}{6}\)

    \(\sqrt{3}\)

    \( \tan \; \frac{7π}{4}\)

    \( \sec \; \frac{3π}{4}\)

    \(−\sqrt{2}\)

    \( \csc \; \csc \; \frac{5π}{4}\)

    \( \cot \; \frac{11π}{4}\)

    −1

    \( \tan \; \frac{8π}{3}\)

    \( \sec \; \frac{4π}{3}\)

    −2

    \( \csc \; \csc \; \frac{2π}{3}\)

    \( \cot \; \frac{5π}{3}\)

    \(−\frac{\sqrt{3}}{3}\)

    \( \tan \; 225°\)

    \( \sec \; 300°\)

    2

    \( \csc \;150°\)csc \;150°\)

    \( \cot \; 240°\)

    \(\frac{\sqrt{3}}{3}\)

    \( \tan \; 330°\)

    \( \sec \; 120°\)

    −2

    \( \csc \; 210°\)csc \; 210°\)

    \( \cot \; 315°\)

    −1

    If \( \sin t= \frac{3}{4}\), and \(t\) is in quadrant II, find \( \cos t, \sec t, \csc t, \tan t, \cot t \).csc t, \tan t, \cot t \).

    If \( \cos t=−\frac{1}{3},\) and \(t\) is in quadrant III, find \( \sin t, \sec t, \csc t, \tan t, \cot t.\)csc t, \tan t, \cot t.\)

    If \(\sin t=−\frac{2\sqrt{2}}{3}, \sec t=−3, \csc t=−\csc t=−\frac{3\sqrt{2}}{4},\tan t=2\sqrt{2}, \cot t= \frac{\sqrt{2}}{4}\)

    If \(\tan t=\frac{12}{5},\) and \(0≤t< \frac{π}{2}\),  find \( \sin t, \cos t, \sec t, \csc t,\) and \(\cot t\).csc t,\) and \(\cot t\).

    If \( \sin t= \frac{\sqrt{3}}{2}\) and \( \cos t=\frac{1}{2},\) find \( \sec t, \csc t, \tan t,\) and  \( \cot t.\)csc t, \tan t,\) and  \( \cot t.\)

    \( \sec t=2, \csc t=\csc t=\frac{2\sqrt{3}}{3}, \tan t= \sqrt{3}, \cot t= \frac{\sqrt{3}}{3}\)

    If \( \sin 40°≈0.643 \; \cos 40°≈0.766 \; \sec 40°,\csc 40°,\tan 40°, \text{ and } \cot 40°.\)csc 40°,\tan 40°, \text{ and } \cot 40°.\)

    If \( \sin t= \frac{\sqrt{2}}{2},\) what is the \( \sin (−t)?\)

    \(−\frac{\sqrt{2}}{2}\)

    If \( \cos t= \frac{1}{2},\) what is the \( \cos (−t)?\)

    If \( \sec t=3.1,\) what is the \( \sec (−t)?\)

    3.1

    If \( \csc t=0.34,\) what is the \( \csc (−t)?\)csc t=0.34,\) what is the \( \csc (−t)?\)

    If \( \tan t=−1.4,\) what is the \( \tan (−t)?\)

    1.4

    If \( \cot t=9.23,\) what is the \( \cot (−t)?\)

    Graphical

    For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

     

     

    \( \sin t= \frac{\sqrt{2}}{2}, \cos t= \frac{\sqrt{2}}{2}, \tan t=1,\cot t=1,\sec t= \sqrt{2}, \csc t= \csc t= \sqrt{2} \)

     

     

     

    \( \sin t=−\frac{\sqrt{3}}{2}, \cos t=−\frac{1}{2}, \tan t=\sqrt{3}, \cot t= \frac{\sqrt{3}}{3}, \sec t=−2, \csc t=−\csc t=−\frac{2\sqrt{3}}{3} \)

     

    Technology

    For the following exercises, use a graphing calculator to evaluate.

    \( \csc \csc \frac{5π}{9}\)

    \( \cot \frac{4π}{7}\)

    –0.228

    \( \sec \frac{π}{10}\)

    \( \tan \frac{5π}{8}\)

    –2.414

    \( \sec \frac{3π}{4}\)

    \( \csc \csc \frac{π}{4}\)

    1.414

    \( \tan 98°\)

    \( \cot 33°\)

    1.540

    \( \cot 140°\)

    \( \sec 310° \)

    1.556

     

    Extensions

    For the following exercises, use identities to evaluate the expression.

    If \(\tan (t)≈2.7,\) and \( \sin (t)≈0.94,\) find \( \cos (t)\).

    If \( \tan (t)≈1.3,\) and \( \cos (t)≈0.61\), find \( \sin (t)\).

    \( \sin (t)≈0.79 \)

    If \( \csc (t)≈3.2,\) and \( \csc (t)≈3.2,\) and \( \cos (t)≈0.95,\) find \( \tan (t)\).

    If \( \cot (t)≈0.58,\) and \( \cos (t)≈0.5,\) find \( \csc (t)\).csc (t)\).

    \( \csc t≈1.16\)csc t≈1.16\)

    Determine whether the function \(f(x)=2 \sin x \cos x\) is even, odd, or neither.

    Determine whether the function \(f(x)=3 \sin ^2 x \cos x + \sec x\) is even, odd, or neither.

    even

    Determine whether the function \(f(x)= \sin x −2 \cos ^2 x \) is even, odd, or neither.

    Determine whether the function \(f(x)= \csc ^2 x+ \sec x\) is even, odd, or neither.csc ^2 x+ \sec x\) is even, odd, or neither.

    even

    For the following exercises, use identities to simplify the expression.

    \( \csc t \tan t\)csc t \tan t\)

    \( \frac{\sec t}{ \csc t}\)csc t}\)

    \( \frac{ \sin t}{ \cos t}= \tan t\)

     

    Real-World Applications

    The amount of sunlight in a certain city can be modeled by the function \(h=15 \cos (\frac{1}{600}d),\) where \(h\) represents the hours of sunlight, and \(d\) is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42nd day of the year. State the period of the function.

    The amount of sunlight in a certain city can be modeled by the function \(h=16 \cos (\frac{1}{500}d)\), where \(h\)  represents the hours of sunlight, and \(d\)  is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267th day of the year. State the period of the function.

    13.77 hours, period: \(1000π\)

    The equation \(P=20 \sin (2πt)+100\) models the blood pressure, \(P\), where \(t\)  represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

    The height of a piston, \(h\), in inches, can be modeled by the equation \(y=2 \cos x+6,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55°\).

    7.73 inches

    The height of a piston, \(h\),in inches, can be modeled by the equation \(y=2 \cos x+5,\) where \(x\) represents the crank angle. Find the height of the piston when the crank angle is \(55°\).

    Review Exercises

    For the following exercises, find the exact value of the given expression.

    \( \cos \frac{π}{6} \)

    \( \tan \frac{π}{4} \)

    1

    \( \csc \frac{π}{3}\)

    \( \sec \frac{π}{4} \)

    \(\sqrt{2}\)

    For the following exercises, use reference angles to evaluate the given expression.

    \( \sec \frac{11π}{3}\)

    \( \sec 315°\)

    \( \sqrt{2}\)

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

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

    0.6

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

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

    \(\frac{\sqrt{2}}{2}\) or \(−\frac{\sqrt{2}}{2}\)

    Which trigonometric functions are even?

    Which trigonometric functions are odd?

    sine, cosecant, tangent, cotangent

    5.4: Right Triangle Trigonometry

    Section Exercises

    Verbal

    For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

     

    When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the x- and y-coordinates?

    The tangent of an angle compares which sides of the right triangle?

    The tangent of an angle is the ratio of the opposite side to the adjacent side.

    What is the relationship between the two acute angles in a right triangle?

    Explain the cofunction identity.

    For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

    Algebraic

    For the following exercises, use cofunctions of complementary angles.

    \( \cos (34°)= \sin (\_\_°)\)

    \( \cos (\frac{π}{3})= \sin (\_\_\_) \)

    \(\frac{π}{6}\)

    \( \csc (21°) = \sec (\_\_\_°)\)csc (21°) = \sec (\_\_\_°)\)

    \( \tan (\frac{π}{4})= \cot (\_\_)\)

    \(\frac{π}{4}\)

    For the following exercises, find the lengths of the missing sides if side \(a\) is opposite angle \(A\), side \(b\)  is opposite angle \(B\), and side \(c\) is the hypotenuse.

    \( \cos B= \frac{4}{5},a=10\)

    \( \sin B= \frac{1}{2}, a=20\)

    \(b= \frac{20\sqrt{3}}{3},c= \frac{40\sqrt{3}}{3}\)

    \( \tan A= \frac{5}{12},b=6\)

    \( \tan A=100,b=100\)

    \(a=10,000,c=10,000.5\)

    \(\sin B=\frac{1}{\sqrt{3}}, a=2 \)

    \(a=5, ∡ A=60^∘\)

    \(b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}\)

    \(c=12, ∡ A=45^∘\)

    Graphical

    For the following exercises, use Figure to evaluate each trigonometric function of angle A.

    \(\sin A\)

    \(\frac{5\sqrt{29}}{29}\)

    \( \cos A \)

    \( \tan A \)

    \(\frac{5}{2}\)

    \(\csc A \)csc A \)

    \( \sec A \)

    \(\frac{\sqrt{29}}{2}\)

    \( \cot A \)

    For the following exercises, use Figure to evaluate each trigonometric function of angle A.

    \( \sin A\)

    \(\frac{5\sqrt{41}}{41}\)

    \( \cos A\)

    \( \tan A \)

    \(\frac{5}{4}\)

    \( \csc A\)csc A\)

    \( \sec A\)

    \(\frac{\sqrt{41}}{4}\)

    \(\cot A\)

    For the following exercises, solve for the unknown sides of the given triangle.

     

    \(c=14, b=7\sqrt{3}\)

     

    \(a=15, b=15 \)

     

    Technology

    For the following exercises, use a calculator to find the length of each side to four decimal places.


    \(b=9.9970, c=12.2041\)

     

    \(a=2.0838, b=11.8177\)

    \(b=15, ∡B=15^∘\)

    \(a=55.9808,c=57.9555\)

    \(c=200, ∡B=5^∘\)

    \(c=50, ∡B=21^∘\)

    \(a=46.6790,b=17.9184\)

    \(a=30, ∡A=27^∘\)

    \(b=3.5, ∡A=78^∘\)

    \(a=16.4662,c=16.8341\)

    Extensions

    Find \(x\).

     

    Find \(x\).

    188.3159

     

    Find \(x\).

     

    Find \(x\).

    200.6737

    A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(36°\), and that the angle of depression to the bottom of the tower is \(23°\). How tall is the tower?

    A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is \(43°\), and that the angle of depression to the bottom of the tower is \(31°\). How tall is the tower?

    498.3471 ft

    A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(15°\), and that the angle of depression to the bottom of the tower is \(2°\). How far is the person from the monument?

    A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is \(18°\), and that the angle of depression to the bottom of the monument is \(3°\). How far is the person from the monument?

    1060.09 ft

    There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be \(40°\). From the same location, the angle of elevation to the top of the antenna is measured to be \(43°\). Find the height of the antenna.

    There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be \(36°\). From the same location, the angle of elevation to the top of the lightning rod is measured to be \(38°\). Find the height of the lightning rod.

    27.372 ft

    Real-World Applications

    A 33-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?  

    A 23-ft ladder leans against a building so that the angle between the ground and the ladder is \(80°\). How high does the ladder reach up the side of the building?

    22.6506 ft

    The angle of elevation to the top of a building in New York is found to be 9 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.

    The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

    368.7633 ft

    Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be \(60°\), how far from the base of the tree am I?

    Review Exercises

    For the following exercises, use side lengths to evaluate.

    \( \cos \frac{π}{4}\)

    \( \cot \frac{π}{3}\)

    \(\frac{\sqrt{3}}{3}\)

    \( \tan \frac{π}{6}\)

    \( \cos (\frac{π}{2}) = \sin ( \_\_°)\)

    0

    \( \csc (18°)= \sec (\_\_°)\)

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

    \( \cos B= \frac{3}{5}, a=6\)

    \( b=8,c=10\)

    \( \tan A = \frac{5}{9},b=6 \)

    For the following exercises, use Figure to evaluate each trigonometric function.

    \( \sin A \)

    \( \frac{11\sqrt{157}}{157}\)

    \( \tan B \)

    For the following exercises, solve for the unknown sides of the given triangle.

     

    \(a=4, b=4 \)

    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?

    14.0954 ft

    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

    Convert \( \frac{5π}{6}\) radians to degrees.

    \(150°\)

    Convert \(−620°\) to radians.

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

    6.283 centimeters

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

    Find the angle between \(0°\) and \(360°\) that is coterminal with \(375°\).

    \(15°\)

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

    Draw the angle \(315°\) in standard position on the Cartesian plane.

    Draw the angle \(−\frac{π}{6}\) in standard position on the Cartesian plane.

    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?

    3.351 feet per second, \( \frac{2π}{75}\) radians per second

    Find the exact value of \( \sin \frac{π}{6}\).

    Compute sine of \(240°\).

    \(−\frac{\sqrt{3}}{2}\)

    State the domain of the sine and cosine functions.

    State the range of the sine and cosine functions.

    \([ –1,1 ]\)

    Find the exact value of \( \cot \frac{π}{4}\).

    Find the exact value of \( \tan \frac{π}{3}\).

    \( \sqrt{3}\)

    Use reference angles to evaluate \( \csc \frac{7π}{4}\).

    Use reference angles to evaluate \( \tan 210°\).

    \(\frac{\sqrt{3}}{3}\)

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

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

    \(\frac{\sqrt{3}}{2}\)

    Which trigonometric functions are even?

    Find the missing angle: \( \cos (\frac{π}{6})= \sin (\_\_\_) \)

    \(\frac{π}{3}\)

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

    Find the missing sides of the triangle.

    \(a=\frac{9}{2},b=\frac{9\sqrt{3}}{2}\)

    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.