Right Triangle Trigonometry: Learn It 1

  • Use right triangles to evaluate trigonometric functions.
  • Use cofunctions of complementary angles.
  • Use the definitions of trigonometric functions of any angle.
  • Use right triangle trigonometry to solve applied problems.

Using Right Triangles to Evaluate Trigonometric Functions

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 [latex]t[/latex] is its value at [latex]t[/latex] radians. First, we need to create our right triangle. If we drop a vertical line segment from the point [latex](x,y)[/latex] to the x-axis, we have a right triangle whose vertical side has length [latex]y[/latex] and whose horizontal side has length [latex]x[/latex]. We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.

 

Graph of quarter circle with radius of 1 and angle of t. Point of (x,y) is at intersection of terminal side of angle and edge of circle.

[latex]\cos t=x[/latex] and [latex]\sin t=y[/latex]

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 [latex]\left(x,y\right)[/latex] coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of [latex]x[/latex], we will call the side between the given angle and the right angle the adjacent side to angle [latex]t[/latex]. (Adjacent means “next to.”) Instead of [latex]y[/latex], we will call the side most distant from the given angle the opposite side from angle [latex]t[/latex]. And instead of [latex]1[/latex], we will call the side of a right triangle opposite the right angle the hypotenuse.

labeling right triangle sides

A right triangle with hypotenuse, opposite, and adjacent sides labeled.

The sides of a right triangle in relation to angle [latex]t[/latex].

Understanding Right Triangle Relationships

Given a right triangle with an acute angle of [latex]t[/latex],

[latex]\begin{align}&\sin \left(t\right)=\frac{\text{opposite}}{\text{hypotenuse}} \\ &\cos \left(t\right)=\frac{\text{adjacent}}{\text{hypotenuse}} \\ &\tan \left(t\right)=\frac{\text{opposite}}{\text{adjacent}} \end{align}[/latex]

A common mnemonic for remembering these relationships is SOH-CAH-TOA, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

How To: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle.

  1. Find the sine as the ratio of the opposite side to the hypotenuse.
  2. Find the cosine as the ratio of the adjacent side to the hypotenuse.
  3. Find the tangent is the ratio of the opposite side to the adjacent side.
Given the triangle, find the value of [latex]\cos \alpha[/latex].
A right triangle with sid lengths of 8, 15, and 17. Angle alpha also labeled.

Given the triangle, find the value of [latex]\text{sin}t[/latex].

A right triangle with sides of 7, 24, and 25. Also labeled is angle t.