Problem 95 - Algebra and Trigonometry

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 the sine of \( \frac{\pi}{4} \) rad (or 45 degrees), can be determined using an isosceles right triangle and is \( \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \). Also, using a calculator set in radian mode would yield the same result.

Step 1: Understanding the Statement

First, try to understand the statement: '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.' This statement seems to make sense as you can indeed use an isosceles right triangle to determine the exact value of \( \sin \frac{\pi}{4} \), and also use a calculator for the same.

Step 2: Analyzing the Geometrical Aspect

An isosceles right triangle has a 90-degree angle, and the other two angles are each 45 degrees or \( \frac{\pi}{4} \) in radians. Since sine of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse, and in an isosceles right triangle these lengths are equal, \( \sin \frac{\pi}{4} \) equals 1/√2 or √2/2.

Step 3: Analyzing the Calculator Usage

Indeed, computation using a calculator is the modern way of finding the trigonometric values. If a calculator is used to find the sine of \( \frac{\pi}{4} \), it would also give the value √2/2, assuming the calculator is in radian mode.