1.2 Unit circle: sine and cosine functions  (Page 8/12)

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$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{\sqrt{3}}{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{1}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\frac{\sqrt{2}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{sin}\text{\hspace{0.17em}}\pi$

0

$\mathrm{sin}\text{\hspace{0.17em}}\frac{3\pi }{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\pi$

−1

$\mathrm{cos}\text{\hspace{0.17em}}0$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\frac{\sqrt{3}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}0$

Numeric

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

$240°$

$60°$

$-170°$

$100°$

$80°$

$-315°$

$135°$

$45°$

$\frac{5\pi }{4}$

$\frac{2\pi }{3}$

$\frac{\pi }{3}$

$\frac{5\pi }{6}$

$\frac{-11\pi }{3}$

$\frac{\pi }{3}$

$\frac{-\text{\hspace{0.17em}}7\pi }{4}$

$\frac{-\pi }{8}$

$\frac{\pi }{8}$

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, $\text{sin}\left(300°\right)=-\frac{\sqrt{3}}{2},\mathrm{cos}\left(300°\right)=\frac{1}{2}\text{\hspace{0.17em}}$

$320°$

$135°$

$45°,$ Quadrant II, $\text{\hspace{0.17em}}\text{sin}\left(135°\right)=\frac{\sqrt{2}}{2},$ $\text{\hspace{0.17em}}\mathrm{cos}\left(135°\right)=-\frac{\sqrt{2}}{2}$

$210°$

$120°$

$60°,$ Quadrant II, $\text{\hspace{0.17em}}\text{sin}\left(120°\right)=\frac{\sqrt{3}}{2},$ $\text{\hspace{0.17em}}\mathrm{cos}\left(120°\right)=-\frac{1}{2}$

$250°$

$150°$

$\text{\hspace{0.17em}}30°,$ Quadrant II, $\text{\hspace{0.17em}}\text{sin}\left(150°\right)=\frac{1}{2},$ $\text{\hspace{0.17em}}\mathrm{cos}\left(150°\right)=-\frac{\sqrt{3}}{2}$

$\frac{5\pi }{4}$

$\frac{7\pi }{6}$

$\frac{\pi }{6},$ Quadrant III, $\text{\hspace{0.17em}}\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2},$ $\text{cos}\left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}$

$\frac{5\pi }{3}$

$\frac{3\pi }{4}$

$\frac{\pi }{4},$ Quadrant II, $\text{\hspace{0.17em}}\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2},$ $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{4\pi }{3}\right)=-\frac{\sqrt[]{2}}{2}$

$\frac{4\pi }{3}$

$\frac{2\pi }{3}$

$\frac{\pi }{3},$ Quadrant II, $\text{\hspace{0.17em}}\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2},$ $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{2\pi }{3}\right)=-\frac{1}{2}$

$\frac{5\pi }{6}$

$\frac{7\pi }{4}$

$\frac{\pi }{4},$ Quadrant IV, $\text{\hspace{0.17em}}\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2},$ $\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}$

For the following exercises, find the requested value.

If $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{1}{7}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in the 4 th quadrant, find $\text{\hspace{0.17em}}\text{sin}\left(t\right).$

If $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{2}{9}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in the 1 st quadrant, find $\text{\hspace{0.17em}}\text{sin}\left(t\right).\text{\hspace{0.17em}}$

$\frac{\sqrt{77}}{9}$

If $\text{\hspace{0.17em}}\text{sin}\left(t\right)=\frac{3}{8}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in the 2 nd quadrant, find $\text{\hspace{0.17em}}\text{cos}\left(t\right).$

If $\text{\hspace{0.17em}}\text{sin}\left(t\right)=-\frac{1}{4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in the 3 rd quadrant, find $\text{\hspace{0.17em}}\text{cos}\left(t\right).$

$-\frac{\sqrt{15}}{4}$

Find the coordinates of the point on a circle with radius 15 corresponding to an angle of $\text{\hspace{0.17em}}220°.$

Find the coordinates of the point on a circle with radius 20 corresponding to an angle of $\text{\hspace{0.17em}}120°.$

$\text{\hspace{0.17em}}\left(-10,10\sqrt{3}\right)\text{\hspace{0.17em}}$

Find the coordinates of the point on a circle with radius 8 corresponding to an angle of $\text{\hspace{0.17em}}\frac{7\pi }{4}.$

Find the coordinates of the point on a circle with radius 16 corresponding to an angle of $\text{\hspace{0.17em}}\frac{5\pi }{9}.$

$\text{\hspace{0.17em}}\left(–2.778,15.757\right)\text{\hspace{0.17em}}$

State the domain of the sine and cosine functions.

State the range of the sine and cosine functions.

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[–1,1\right]\text{\hspace{0.17em}}$

Graphical

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}.}$

$\mathrm{sin}t=\frac{1}{2},\mathrm{cos}t=-\frac{\sqrt{3}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{\sqrt{2}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=-\frac{\sqrt{2}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=-\frac{1}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{\sqrt{2}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=0,\mathrm{cos}\text{\hspace{0.17em}}t=-1$

$\mathrm{sin}\text{\hspace{0.17em}}t=-0.596,\mathrm{cos}\text{\hspace{0.17em}}t=0.803$

$\mathrm{sin}\text{\hspace{0.17em}}t=\frac{1}{2},\mathrm{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{1}{2},\mathrm{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=0.761,\mathrm{cos}\text{\hspace{0.17em}}t=-0.649$

$\mathrm{sin}\text{\hspace{0.17em}}t=1,\mathrm{cos}\text{\hspace{0.17em}}t=0$

Technology

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

$\mathrm{sin}\text{\hspace{0.17em}}\frac{5\pi }{9}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{5\pi }{9}$

−0.1736

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{10}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{10}$

0.9511

$\mathrm{sin}\text{\hspace{0.17em}}\frac{3\pi }{4}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{3\pi }{4}$

−0.7071

$\mathrm{sin}\text{\hspace{0.17em}}98°$

$\mathrm{cos}\text{\hspace{0.17em}}98°$

−0.1392

$\mathrm{cos}\text{\hspace{0.17em}}310°$

$\mathrm{sin}\text{\hspace{0.17em}}310°$

−0.7660

Extensions

$\mathrm{sin}\left(\frac{11\pi }{3}\right)\mathrm{cos}\left(\frac{-5\pi }{6}\right)$

$\mathrm{sin}\left(\frac{3\pi }{4}\right)\mathrm{cos}\left(\frac{5\pi }{3}\right)$

$\frac{\sqrt{2}}{4}$

$\mathrm{sin}\left(-\frac{4\pi }{3}\right)\mathrm{cos}\left(\frac{\pi }{2}\right)$

$\mathrm{sin}\left(\frac{-9\pi }{4}\right)\mathrm{cos}\left(\frac{-\pi }{6}\right)$

$-\frac{\sqrt{6}}{4}$

$\mathrm{sin}\left(\frac{\pi }{6}\right)\mathrm{cos}\left(\frac{-\pi }{3}\right)$

$\mathrm{sin}\left(\frac{7\pi }{4}\right)\mathrm{cos}\left(\frac{-2\pi }{3}\right)$

$\frac{\sqrt{2}}{4}$

$\mathrm{cos}\left(\frac{5\pi }{6}\right)\mathrm{cos}\left(\frac{2\pi }{3}\right)$

$\mathrm{cos}\left(\frac{-\pi }{3}\right)\mathrm{cos}\left(\frac{\pi }{4}\right)$

$\frac{\sqrt{2}}{4}$

$\mathrm{sin}\left(\frac{-5\pi }{4}\right)\mathrm{sin}\left(\frac{11\pi }{6}\right)$

$\mathrm{sin}\left(\pi \right)\mathrm{sin}\left(\frac{\pi }{6}\right)$

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 $\text{\hspace{0.17em}}\left(0,1\right),$ 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?

$\left(0,–1\right)$

What is the coordinates of the child after 125 seconds?

When will the child have coordinates $\text{\hspace{0.17em}}\left(0.707,–0.707\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\left(-0.866,-0.5\right)\text{\hspace{0.17em}}$ if the ride last 6 minutes?

Questions & Answers

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Eke Reply
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Almas
are nano particles real
Missy Reply
yeah
Joseph
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Lale Reply
no can't
Lohitha
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William
currently
William
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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Maira Reply
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learn
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learn
Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
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Crow Reply
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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Mahi
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Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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