Chapter 4.3, Problem 53E

Solution

To state: The value of expressions in exercise 49-52 by using a calculator. Explain the answer if it is correct or not.

The calculation using the calculator is wrong. It happened because the value of $\pi$ stored in calculators is correct only up to few decimals. When this value is multiplied with large numbers this error gets amplified and causes error in final answer.

Given information:

The given trigonometric functions are:

  $\sin \left(\frac{\pi }{6}+49,000\pi \right),\tan \left(1,234,567\pi \right)-\tan \left(7,654,321\pi \right),\cos \left(\frac{5,555,555\pi }{2}\right),\tan \left(\frac{3\pi -70,000\pi }{2}\right)$

Explanation:

Consider the given function: $\sin \left(\frac{\pi }{6}+49,000\pi \right)$

Use the Ti-calculator to find the value of the expression $\sin \left(\frac{\pi }{6}+49,000\pi \right)$ , type the expression in the calculator and press ENTER.

  

Therefore, the value of $\sin \left(\frac{\pi }{6}+49,000\pi \right)$ is $0.5000000036$ .

Now consider the function: $\tan \left(1,234,567\pi \right)-\tan \left(7,654,321\pi \right)$

Use the Ti-calculator to find the value of the expression $\tan \left(1,234,567\pi \right)-\tan \left(7,654,321\pi \right)$ , type the expression in the calculator and press ENTER.

  

Therefore, the value of $\tan \left(1,234,567\pi \right)-\tan \left(7,654,321\pi \right)$ is $0.0000001329092$ .

Now consider the function: $\cos \left(\frac{5,555,555\pi }{2}\right)$

Use the Ti-calculator to find the value of the expression $\cos \left(\frac{5,555,555\pi }{2}\right)$ , type the expression in the calculator and press ENTER.

  

Therefore, the value of $\cos \left(\frac{5,555,555\pi }{2}\right)$ is $0$ .

Now consider the function: $\tan \left(\frac{3\pi -70,000\pi }{2}\right)$

Use the Ti-calculator to find the value of the expression $\tan \left(\frac{3\pi -70,000\pi }{2}\right)$ , type the expression in the calculator and press ENTER.

  

Therefore, the value of $\tan \left(\frac{3\pi -70,000\pi }{2}\right)$ is $135415115$ .

From the above calculations, it can be clearly seen that the above calculation in wrong. It happened because the value of $\pi$ stored in calculators is correct only up to few decimals. When this value is multiplied with large numbers this error gets amplified and causes error in final answer.