Questions: Suppose cos(A) = 0.325 Use the trig identity sin^2(A) + cos^2(A) = 1 and the trig identity tan(A) = sin(A) / cos(A) to find tan(A) in quadrant IV. Round to ten-thousandth.

Suppose cos(A) = 0.325

Use the trig identity sin^2(A) + cos^2(A) = 1 and the trig identity tan(A) = sin(A) / cos(A) to find tan(A) in quadrant IV. Round to ten-thousandth.

Transcript text: Suppose $\cos (A)=0.325$ Use the trig identity $\sin ^{2}(A)+\cos ^{2}(A)=1$ and the trig identity $\tan (A)=\frac{\sin (A)}{\cos (A)}$ to find $\tan (A)$ in quadrant IV. Round to ten-thousandth.

Solution

Solution Steps

To solve this problem, we need to find the value of $\tan(A)$ given that $\cos(A) = 0.325$ and $A$ is in the fourth quadrant. We will use the trigonometric identity $\sin^2(A) + \cos^2(A) = 1$ to find $\sin(A)$. Since $A$ is in the fourth quadrant, $\sin(A)$ will be negative. Once we have $\sin(A)$, we can calculate $\tan(A)$ using the formula $\tan(A) = \frac{\sin(A)}{\cos(A)}$.

Step 1: Given Information

We are given that $ \cos(A) = 0.325 $ and we need to find $ \tan(A) $ in the fourth quadrant.

Step 2: Calculate $ \sin(A) $

Using the trigonometric identity $ \sin^2(A) + \cos^2(A) = 1 $, we can find $ \sin(A) $: \[ \sin^2(A) = 1 - \cos^2(A) = 1 - (0.325)^2 = 1 - 0.105625 = 0.894375 \] Taking the square root, we find: \[ \sin(A) = -\sqrt{0.894375} \approx -0.9457 \] (Note: We take the negative root because $ A $ is in the fourth quadrant.)

Step 3: Calculate $ \tan(A) $

Now, we can calculate $ \tan(A) $ using the formula: \[ \tan(A) = \frac{\sin(A)}{\cos(A)} = \frac{-0.9457}{0.325} \approx -2.9099 \]