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We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
In this section, we will see another way to define trigonometric functions using properties of right triangles .
In earlier sections, we used a unit circle to define the trigonometric functions . In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is its value at $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ radians. First, we need to create our right triangle. [link] shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point $\text{\hspace{0.17em}}(x,y)\text{\hspace{0.17em}}$ to the x -axis, we have a right triangle whose vertical side has length $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ and whose horizontal side has length $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.
We know
Likewise, we know
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using $\text{\hspace{0.17em}}(x,y)\text{\hspace{0.17em}}$ coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of $\text{\hspace{0.17em}}x,$ we will call the side between the given angle and the right angle the adjacent side to angle $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ (Adjacent means “next to.”) Instead of $\text{\hspace{0.17em}}y,$ we will call the side most distant from the given angle the opposite side from angle $\text{}t.\text{\hspace{0.17em}}$ And instead of $\text{\hspace{0.17em}}1,$ we will call the side of a right triangle opposite the right angle the hypotenuse . These sides are labeled in [link] .
Given a right triangle with an acute angle of $\text{\hspace{0.17em}}t,$
A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ S ine is o pposite over h ypotenuse, C osine is a djacent over h ypotenuse, T angent is o pposite over a djacent.”
Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.
Given the triangle shown in [link] , find the value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha .$
The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so:
Given the triangle shown in [link] , find the value of $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t.$
$\frac{7}{25}$
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