How to find adjacent side with hypotenuse and opposite?

Understand the Problem

The question is asking how to calculate the length of the adjacent side of a right triangle when the lengths of the hypotenuse and the opposite side are known. To solve this, we can use trigonometric ratios, specifically the sine and cosine functions.

Answer

The length of the adjacent side is $ a = h \cdot \cos(\theta) $.

Steps to Solve

Identify the relevant triangle function Since we need to find the length of the adjacent side ($a$) in a right triangle, and we know the hypotenuse ($h$) and the opposite side ($o$), we should use the cosine function because it relates the adjacent side to the hypotenuse.
Write the cosine ratio formula The cosine of an angle in a right triangle is defined as the adjacent side divided by the hypotenuse: $$ \cos(\theta) = \frac{a}{h} $$
Rearrange the formula to find the adjacent side To find the length of the adjacent side, rearrange the equation to solve for $a$: $$ a = h \cdot \cos(\theta) $$
Substitute known values Once we have the appropriate values of the hypotenuse $h$ and angle $\theta$, we can substitute them into our equation to find $a$: For example: $$ a = h \cdot \cos(\theta) $$
Calculate the result Perform the multiplication to get the length of the adjacent side.

The length of the adjacent side is given by the formula ( a = h \cdot \cos(\theta) ).

More Information

In a right triangle, the cosine function is a key relationship in trigonometry. It helps to find the lengths of sides when angles and one side length are known. Understanding these relationships is crucial in geometry and trigonometry.

Tips

Mixing up the sine and cosine functions. Remember, sine is for the opposite side and cosine is for the adjacent side.
Not converting angles to radians when using calculators set to radians.
Forgetting to use the right angle in the calculations can lead to incorrect results.

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