Problem 48 - Precalculus: A Concise Course

Question

Use the value of the trigonometric function to evaluate the indicated functions. \(\cos t=\frac{4}{5}\) (a) \(\cos (\pi-t)\) (b) \(\cos (t+\pi)\)

Step-by-Step Solution

Answer

The solution for \(\cos (\pi-t)\) is -\(\frac{4}{5}\) and the solution for \(\cos (t+\pi)\) is -\(\frac{4}{5}\).

Step 1: Understand cosine function properties

An even function is symmetrical about the y-axis. The cosine function is an even function because the cosine of an angle equals the cosine of the negative of that angle, or \(\cos (-x) = \cos x\).

Step 2: Solve for \(\cos (\pi-t)\)

Using the property \(\cos (-x) = \cos x\) we can translate \(\cos (\pi-t)\) into \(\cos (t-\pi)\), which then becomes \(\cos t \cdot \cos \pi + \sin t \cdot \sin \pi\). Since the cos t equals to \(\frac{4}{5}\) and \(\cos \pi = -1\), and \(\sin \pi = 0\), the expression simplifies to \(\frac{4}{5} \cdot -1 + 0 = -\frac{4}{5}\).

Step 3: Solve for \(\cos (t+\pi)\)

Using the property \(\cos x = -\cos (x+\pi)\) where \(x = t\), we substitute as follows: \(\cos (t+\pi) = -\cos t\). Since we know that \(\cos t = \frac{4}{5}\), the expression simplifies to \(-\frac{4}{5}\).