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
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\(45°\)
2) \(−\dfrac{5π}{3}\)
For the exercises 3-6, convert the angle measures to radians.
3) \(-210°\)
- Answer
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\(−\dfrac{7π}{6}\)
4) \(180°\)
5) Find the length of an arc in a circle of radius \(7\) meters subtended by the central angle of \(85°\).
- Answer
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\(10.385\) 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
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\(60°\)
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
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\(\dfrac{2π}{11}\)
10) \(\dfrac{14π}{5}\)
For the exercises 11-, draw the angle provided in standard position on the Cartesian plane.
11) \(-210°\)
- Answer
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12) \(75°\)
13) \(\dfrac{5π}{4}\)
- Answer
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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
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\(1036.73\) 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
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\(\dfrac{\sqrt{3}}{2}\)
2) Find the exact value of \( \cos \dfrac{π}{4}\).
3) Find the exact value of \( \cos π \).
- Answer
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\(-1\)
4) State the reference angle for \(300°\).
5) State the reference angle for \( \dfrac{3π}{4}\).
- Answer
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\( \dfrac{π}{4}\)
6) Compute cosine of \(330°\).
7) Compute sine of \(\dfrac{5π}{4}\).
- Answer
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\(−\dfrac{\sqrt{2}}{2}\)
8) State the domain of the sine and cosine functions.
9) State the range of the sine and cosine functions.
- Answer
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\([–1,1]\)
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
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\(1\)
3) \( \csc \dfrac{π}{3}\)
4) \( \sec \dfrac{π}{4} \)
- Answer
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\(\sqrt{2}\)
For the exercises 4-12, use reference angles to evaluate the given expression.
5) \( \sec \dfrac{11π}{3}\)
6) \( \sec 315°\)
- Answer
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\( \sqrt{2}\)
7) If \( \sec (t)=−2.5\), what is the \( \sec (−t)\)?
8) If \( \tan (t)=−0.6 \), what is the \( \tan (−t)\)?
- Answer
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\(0.6\)
9) If \( \tan (t)=\dfrac{1}{3}\), find \( \tan (t−π)\).
10) If \( \cos (t)= \dfrac{\sqrt{2}}{2}\), find \( \sin (t+2π)\).
- Answer
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\(\dfrac{\sqrt{2}}{2}\) or \(−\dfrac{\sqrt{2}}{2}\)
11) Which trigonometric functions are even?
12) Which trigonometric functions are odd?
- Answer
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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
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\(\dfrac{\sqrt{3}}{3}\)
3) \( \tan \dfrac{π}{6}\)
4) \( \cos (\dfrac{π}{2}) = \sin ( \_\_°)\)
- Answer
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\(0\)
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
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\( b=8,c=10\)
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
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\( \dfrac{11\sqrt{157}}{157}\)
9) \( \tan B \)
For the exercises 10-11, solve for the unknown sides of the given triangle.
10)
- Answer
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\(a=4, b=4 \)
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
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\(14.0954\) 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
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\(150°\)
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
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\(6.283\) 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
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\(15°\)
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
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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
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\(3.351\) 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
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\(−\dfrac{\sqrt{3}}{2}\)
12) State the domain of the sine and cosine functions.
13) State the range of the sine and cosine functions.
- Answer
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\([ –1,1 ]\)
14) Find the exact value of \( \cot \dfrac{π}{4}\).
15) Find the exact value of \( \tan \dfrac{π}{3}\).
- Answer
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\( \sqrt{3}\)
16) Use reference angles to evaluate \( \csc \dfrac{7π}{4}\).
17) Use reference angles to evaluate \( \tan 210°\).
- Answer
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\(\dfrac{\sqrt{3}}{3}\)
18) If \( \csc t=0.68\), what is the \( \csc (−t)\)?
19) If \( \cos t= \dfrac{\sqrt{3}}{2}\), find \( \cos (t−2π)\).
- Answer
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\(\dfrac{\sqrt{3}}{2}\)
20) Which trigonometric functions are even?
21) Find the missing angle: \(\cos \left(\dfrac{\pi }{6} \right)= \sin (\;)\)
- Answer
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\(\dfrac{π}{3}\)
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
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\(a=\dfrac{9}{2},b=\dfrac{9\sqrt{3}}{2}\)
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.