Problem 81 - Algebra and Trigonometry

Question

If you are given the lengths of the sides of a right triangle, describe how to find the sine of either acute angle.

Step-by-Step Solution

Answer

The sine of either acute angle in a right triangle can be found by taking the ratio of the length of the side opposite the acute angle to the length of the hypotenuse.

Step 1: Identify the sides of right triangle

First, identify the sides of the right triangle. For any angle in the triangle other than the right angle, the hypotenuse is the longest side (or the side opposite the right angle), the opposite side is the side across from the angle, and the adjacent side is the side next to the angle, i.e., the side which along with the angle forms the right angle.

Step 2: Apply the Sine definition

Next, apply the definition of sine. The sine of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. So, if \(a\), \(b\), and \(c\) are lengths of the hypotenuse, opposite side, and adjacent side respectively, then for the angle \(A\), the sine would be \( \sin A = \frac{b}{a}\) and for the angle \(B\), the sine would be \( \sin B = \frac{c}{a}\) (as \(b\) and \(c\) would be opposite sides for \(A\) and \(B\) respectively).

Step 3: Calculation

Now, calculate the sine of the acute angle using the derived ratio. Put the values of length of sides in the formula derived in step 2 to get the values. Use a calculator if needed to get the ratio in decimal form.