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If $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is a real number and a point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the unit circle corresponds to an angle of $\text{\hspace{0.17em}}t,$ then
Given a point P $\text{\hspace{0.17em}}(x,y)\text{\hspace{0.17em}}$ on the unit circle corresponding to an angle of $\text{\hspace{0.17em}}t,$ find the sine and cosine.
Point $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ is a point on the unit circle corresponding to an angle of $\text{\hspace{0.17em}}t,$ as shown in [link] . Find $\text{\hspace{0.17em}}\mathrm{cos}(t)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{sin}(t).\text{\hspace{0.17em}}$
We know that $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the x -coordinate of the corresponding point on the unit circle and $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the y -coordinate of the corresponding point on the unit circle. So:
A certain angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ corresponds to a point on the unit circle at $\text{\hspace{0.17em}}\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\text{\hspace{0.17em}}$ as shown in [link] . Find $\mathrm{cos}\text{\hspace{0.17em}}t$ and $\mathrm{sin}\text{\hspace{0.17em}}t.$
$\mathrm{cos}(t)=-\frac{\sqrt{2}}{2},\mathrm{sin}(t)=\frac{\sqrt{2}}{2}$
For quadrantral angles, the corresponding point on the unit circle falls on the x- or y -axis. In that case, we can easily calculate cosine and sine from the values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$
Find $\text{\hspace{0.17em}}\mathrm{cos}(\mathrm{90\xb0})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{sin}(\mathrm{90\xb0}).\text{\hspace{0.17em}}$
Moving $\text{\hspace{0.17em}}\mathrm{90\xb0}\text{\hspace{0.17em}}$ counterclockwise around the unit circle from the positive x -axis brings us to the top of the circle, where the $\text{\hspace{0.17em}}(x,y)\text{\hspace{0.17em}}$ coordinates are (0, 1), as shown in [link] .
Using our definitions of cosine and sine,
The cosine of 90° is 0; the sine of 90° is 1.
Find cosine and sine of the angle $\text{\hspace{0.17em}}\pi .\text{\hspace{0.17em}}$
$\text{\hspace{0.17em}}\mathrm{cos}(\pi )=-1,$ $\mathrm{sin}(\pi )=0\text{\hspace{0.17em}}$
Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is $\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=1.\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}x=\mathrm{cos}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}t,$ we can substitute for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ to get $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1.\text{\hspace{0.17em}}$ This equation, $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1,$ is known as the Pythagorean Identity . See [link] .
We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.
The Pythagorean Identity states that, for any real number $\text{\hspace{0.17em}}t,$
Given the sine of some angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and its quadrant location, find the cosine of $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$
If $\text{\hspace{0.17em}}\mathrm{sin}(t)=\frac{3}{7}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in the second quadrant, find $\text{\hspace{0.17em}}\mathrm{cos}(t).\text{\hspace{0.17em}}$
If we drop a vertical line from the point on the unit circle corresponding to $\text{\hspace{0.17em}}t,$ we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See [link] .
Substituting the known value for sine into the Pythagorean Identity,
Because the angle is in the second quadrant, we know the x- value is a negative real number, so the cosine is also negative. So $$\text{cos}(t)=-\frac{2\sqrt{10}}{7}$$
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