Problem 71 - Precalculus Essentials

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although I can use an isosceles right triangle to determine the exact value of \(\sin \frac{\pi}{4},\) I can also use my calculator to obtain this value.

Step-by-Step Solution

Answer

The statement makes sense as both the methods described, that is, using an isosceles right triangle's properties and using a calculator, can accurately compute the value of \(\sin \frac{\pi}{4}\).

Step 1: Understanding the statement

Identify the part of the problem that needs to be addressed - which is to determine if the statement given is making sense or not. In this case, the statement is suggesting two methods to find the value of \(\sin \frac{\pi}{4}\). Also recall the properties of an isosceles right triangle which state that the triangle has two angles measuring 45 degrees (\(\frac{\pi}{4}\) in radians) and the sides opposite these angles are of equal length.

Step 2: Analyzing the first method

Consider the first method proposed by the statement – using an isosceles right triangle. Recalling that in an isosceles right triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse equals the sine of the angle. So, \(\sin(\angle \text{in triangle}) = \frac{\text{opposite side}}{\text{hypotenuse}}\). When the angle in question is 45 degrees or \(\frac{\pi}{4}\) radians, due to the properties of isosceles right triangle where the angles are the same, the lengths of the sides opposite these angles are also the same. Therefore, \(\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\). Thus, the first method makes sense.

Step 3: Analyzing the second method

Consider the second method, which is using a calculator to get the sine value. Calculator is designed to compute trigonometric functions, including sine function. So, by inputting \(\frac{\pi}{4}\) or 45 degrees into a calculator, it would provide back the sine value which is again \(\frac{1}{\sqrt{2}}\). Therefore, the second method is also making sense.