Give your answers in exact form. Do NOT use decimals. Type ‘pi’ if you need to use \(\pi\text{.}\)
The radian measure of an angle of \(264\) degrees is
Answer.
\(\frac{264\pi }{180}\)
Section 5.8 Exercises
These are the exercises in Trigonometry Module.
Checkpoint 5.8.1.
Checkpoint 5.8.2.
What angle (in degrees) corresponds to -3 rotations around the unit circle?
-3 rotations is an angle of degrees.
Answer.
\(-1080\)
Solution.
SOLUTION If we go around -3 times, we make 3 full circles in the negative direction, which is \(-3\cdot 360^{\circ} = -1080^{\circ}\) .
Checkpoint 5.8.3.
Convert each degree measure to radian measure.
Give your answers in exact form. Do NOT use decimals.
Type ’pi’ if you need to use \(\pi\text{.}\)
\(0^\circ\) = radians.
\(360^\circ\) = radians.
\(-330^\circ\) = radians.
\(540^\circ\) = radians.
Answer 1.
\(0\)
Answer 2.
\(6.28318530717959\)
Answer 3.
\(-5.75958653158129\)
Answer 4.
\(9.42477796076938\)
Checkpoint 5.8.4.
From the information given, find the quadrant in which the terminal point determined by \(t\) lies. Input I, II, III, or IV.
(a) \(\sin (t)\lt 0\) and \(\cos (t)\lt 0\text{,}\) quadrant ;
(b) \(\sin (t)>0\) and \(\cos (t)\lt 0\text{,}\) quadrant ;
(c) \(\sin (t)>0\) and \(\cos (t)>0\text{,}\) quadrant ;
(d) \(\sin (t)\lt 0\) and \(\cos (t)>0\text{,}\) quadrant ;
Answer 1.
III
Answer 2.
II
Answer 3.
I
Answer 4.
IV
Checkpoint 5.8.5.
If the point \(P(-\frac{4}{5},y)\) is on the unit circle in quadrant II, then \(y=\) .
Answer.
\(0.6\)
Checkpoint 5.8.6.
For each angle (in radians) below, determine the quadrant in which the terminal side of the angle is found.
[NOTE: Enter ’1’ for quadrant I, ’2’ for quadrant II, ’3’ for quadrant III, and ’4’ for quadrant IV.]
(a) \(\theta = \frac { 8 \pi } { 3 }\) is found in quadrant
(b) \(\theta = \frac { 15 \pi } { 4 }\) is found in quadrant
(c) \(\theta = \frac { 11 \pi } { 6 }\) is found in quadrant
(d) \(\theta = 7\) is found in quadrant
Answer 1.
\(2\)
Answer 2.
\(4\)
Answer 3.
\(4\)
Answer 4.
\(1\)
Checkpoint 5.8.7.
A point \(P\) is often identified in mathematics using the notation \(P(x,y)\) to reference a point at coordinates \((x,y)\text{.}\)
If \(P(x,\frac{8}{10})\) is on the unit circle in quadrant II, then \(x =\) .
Hint.
The equation of the unit circle is
\begin{equation*}
x^2+y^2 = 1
\end{equation*}
If you are given y, you can carefully solve for x.
Watch your algebra!
Answer.
\(-0.6\)
Solution.
SOLUTION:
Evaluating \(x^2+y^2 = 1\) for \(y = \frac{8}{10}\) gives
\begin{equation*}
{(\frac{8}{10}})^2+x^2 = 1
\end{equation*}
Solving for x yields
\begin{equation*}
x^2 = {(0.6)}^2
\end{equation*}
Since the point is in quadrant II, then x is negative and so \(x = -0.6\text{.}\)
Checkpoint 5.8.8.
Find the exact value.
\(\sin(\frac{2}{3}\pi) =\)
\(\cos(\frac{2}{3}\pi) =\)
Answer 1.
\(\frac{\sqrt{3}}{2}\)
Answer 2.
\(\frac{-1}{2}\)
Checkpoint 5.8.9.
For the actue angle \(\theta\) with \(\cot\theta = 1\text{,}\) find
\(\sin\theta =\)
\(\cos\theta =\)
\(\tan\theta =\)
\(\sec\theta =\)
\(\csc\theta =\)
Answer 1.
\(\frac{\sqrt{2}}{2}\)
Answer 2.
\(\frac{\sqrt{2}}{2}\)
Answer 3.
\(1\)
Answer 4.
\(\sqrt{2}\)
Answer 5.
\(\sqrt{2}\)
Checkpoint 5.8.10.
Given \(\tan(\alpha)=\sqrt 3\) and \(\pi\lt \alpha\lt 3\pi/2\text{,}\) find the exact values of the remaining five trigonometric functions.
Note: You are not allowed to use decimals in your answer.
\(\sin(\alpha)\) = .
\(\cos(\alpha)\) = .
\(\csc(\alpha)\) = .
\(\sec(\alpha)\) = .
\(\cot(\alpha)\) = .
Answer 1.
\(-0.866025403784439\)
Answer 2.
\(-0.5\)
Answer 3.
\(-1.15470053837925\)
Answer 4.
\(-2\)
Answer 5.
\(0.577350269189626\)
Checkpoint 5.8.11.
\(\tan (\frac{2\pi}{3})\)
\(\cot (\frac{\pi}{3})\)
\(\sec (0)\)
\(\csc (-\frac{2\pi}{3})\)
Answer 1.
\(-1\)
Answer 2.
\(-0.5\)
Answer 3.
\(-1.73205080756888\)
Answer 4.
\(0.577350269189626\)
Answer 5.
\(1\)
Answer 6.
\(-1.15470053837925\)
Checkpoint 5.8.12.
Find the exact value of each without using a calculator. Remember that you can type “sqrt(a)” if your answer has \(\sqrt{a}\) in it.
\(\tan{\left(\frac{11 \pi}{6}\right)} =\)
\(\tan{\left(\frac{5 \pi}{3}\right)} =\)
\(\cot{\left(\frac{\pi}{3}\right)} =\)
\(\sec{\left(\frac{2 \pi}{3}\right)} =\)
\(\csc{\left(\frac{5 \pi}{4}\right)} =\)
Answer 1.
\(\frac{-1}{\sqrt{3}}\)
Answer 2.
\(-\sqrt{3}\)
Answer 3.
\(\frac{1}{\sqrt{3}}\)
Answer 4.
\(-2\)
Answer 5.
\(-\sqrt{2}\)
Checkpoint 5.8.13.
Simplify and write the trigonometric expression in terms of sine and cosine:
\(\frac{\sec t -\cos t}{\sec t}=(f(t))^2\)
\(f(t)=\)
Answer.
\(\sin\!\left(t\right)\)
Checkpoint 5.8.14.
For each trigonometric expression in the lefthand column, choose the expression from the righthand column that completes a fundamental identity. Enter the appropriate letter (A,B,C,D, or E) in each blank.
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Checkpoint 5.8.15.
Simplify \(\sin(\tan^{-1}{x})\text{.}\)
Answer.
\(\frac{x}{\sqrt{1+x^{2}}}\)
Checkpoint 5.8.16.
Find all solutions of the equation \(\tan^5 x - 9\tan x =0.\)
The answer has the form \(x= Ak\pi\) where \(k=0,\pm 1, \pm 2,\pm 3,\ldots\) ranges over the integers and
the constant \(A=\) .
Answer.
\(0.333333\)
Checkpoint 5.8.17.
Find all solutions of the equation \(\cos x (2\sin x + 1)=0\text{.}\)
The answer is \(A+k\pi\text{,}\) \(B+2k\pi\text{,}\) \(C+2k\pi\) where \(k\) is any integer and \(0\lt A\lt \pi\text{,}\) \(0\lt B\lt C\lt 2\pi\)
\(A =\)
\(B =\)
\(C =\)
Answer 1.
\(\frac{\pi }{2}\)
Answer 2.
\(\frac{7\pi }{6}\)
Answer 3.
\(\frac{11\pi }{6}\)
Checkpoint 5.8.18.
Find all solutions of the equation \(\sec^2 x -2=0\text{.}\)
The answer is \(A+B k\pi\) where \(k\) is any integer and \(0\lt A\lt \frac{\pi}{2}\text{.}\)
\(A =\)
\(B =\)
Answer 1.
\(\frac{\pi }{4}\)
Answer 2.
\(\frac{1}{2}\)
Checkpoint 5.8.19.
Solve the equation in the interval \([0,2\pi]\text{.}\)
Note: The answer must be a multiple of \(\pi\text{.}\) Note that \(\pi\) is already provided in the answer so you must give the appropriate multiple. Give exact answers. No decimal numbers. The asnwer should be a fraction or an integer. E.g. if the solution is \(\frac{\pi}{2}\text{,}\) the answer should be entered as 1/2. If there is more than one answer enter them separated by commas.
\(2\sin t \cos t - \cos t + 2 \sin t-1=0\)
\(t =\)\(\pi\)
Answer.
No correct answer specified
Checkpoint 5.8.20.
For the given angle \(x\) in the triangle below, calculate each of the following.
(We are aware that the image is not displaying and are trying to resolve the issue! The image should show a right triangle whose horizontal leg has length 40, whose vertical leg is not labled, and whose hypotenuse has lenght 41. The angle between the hypotenuse and the vertical leg is labled \(x\text{.}\))
\(\sin x =\)
\(\cos x =\)
\(\cot x =\)
\(\sec x =\)
\(\csc x =\)
Answer 1.
\({\frac{40}{41}}\)
Answer 2.
\({\frac{9}{41}}\)
Answer 3.
\({\frac{40}{9}}\)
Answer 4.
\({\frac{9}{40}}\)
Answer 5.
\({\frac{41}{9}}\)
