Finding the Values of (Theta): Solving cos2theta = 1/4

[ibysaiyan]

Homework Statement

Find the values of (theta),in the interval 0degree$$\leq(theta)$$$$\leq180$$,for which cos2theta=1/4.Give your answers correct to the nearest integers.

Homework Equations

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The Attempt at a Solution

do i just tap them in the calculator or how?

 

[Dick]

I would start by sketching a graph of the function first. You should be able to see that there are two possible values of theta. Then sure, use your calculator to find them.

 

[Architeuthis Dux]

As a scientist, you should approach this problem by using mathematical principles and equations, rather than simply using a calculator. First, recall the double angle identity for cosine: cos2theta = 2cos^2(theta) - 1. Substituting this into the given equation, we have 2cos^2(theta) - 1 = 1/4. Solving for cos^2(theta), we get cos^2(theta) = 5/8. Taking the square root of both sides, we have cos(theta) = ±√(5/8). Since we are looking for solutions in the interval of 0 degrees to 180 degrees, we can disregard the negative solution. Therefore, cos(theta) = √(5/8). Next, we can use the inverse cosine function to find the value of theta that satisfies this equation. Using a calculator, we find that the values of theta that satisfy cos2theta = 1/4 are approximately 49.1 degrees and 130.9 degrees. Rounding to the nearest integers, we have theta = 49 degrees and theta = 131 degrees.