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Chapter 5: Problem 62

Suppose \(-\frac{\pi}{2}<\theta<0\) and \(\cos \theta=0.8\). (a) Without using a double-angle formula, evaluate \(\cos (2 \theta)\) by first finding \(\theta\) using an inverse trigonometric function. (b) Without using an inverse trigonometric function, evaluate \(\cos (2 \theta)\) again by using a double-angle formula.

Short Answer

Expert verified
(a) We found the angle \(\theta\approx-0.6435\) using \(\theta=\arccos(0.8)\), and calculated \(\cos(2\theta)=\cos(-1.287)\approx0.6\). (b) We used the double-angle formula \(\cos(2\theta)=2\cos^2(\theta)-1\), and with \(\cos\theta=0.8\), we found \(\cos(2\theta)=0.28\).

Step by step solution

01

Part (a): Finding \(\theta\) using an inverse trigonometric function

Since \(\cos\theta=0.8\), we can find the angle \(\theta\) using the inverse cosine function. Please note that the given domain for \(\theta\) is \(-\frac{\pi}{2}<\theta<0\): \[\theta=\arccos(0.8).\]
02

Compute the value of \(\theta\)

Now, using a calculator to find the value of the angle: \[\theta\approx-0.6435.\] We have found the angle \(\theta\) in the specified range.
03

Compute the value of \(\cos(2\theta)\)

With the value of \(\theta\) found, we can now calculate the value of \(\cos(2\theta)\): \[\cos(2\theta)=\cos(-1.287)\approx0.6.\] So, the approximate value of \(\cos(2\theta)\) using method (a) is 0.6.
04

Part (b): Finding \(\cos(2\theta)\) using double-angle formula

We will use the double-angle formula for the cosine function, which is: \[\cos(2\theta)=2\cos^2(\theta)-1.\]
05

Substitute the given value of \(\cos\theta\) and compute \(\cos(2\theta)\)

Now, substituting the given value of \(\cos\theta=0.8\) into the formula: \[\cos(2\theta)=2(0.8)^2-1=2(0.64)-1=1.28-1=0.28.\] So, the value of \(\cos(2\theta)\) using method (b) is 0.28. In conclusion, using method (a), we found that \(\cos(2\theta)\approx0.6\). However, using method (b), we found that \(\cos(2\theta)=0.28\).

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