Questions: Find the exact solution of the equation. 3 cos^(-1)(3x) = 2π The solution set is 3. 6 (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed.)

Find the exact solution of the equation.
3 cos^(-1)(3x) = 2π

The solution set is 3. 6 (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed.)

Transcript text: Find the exact solution of the equation. \[ 3 \cos ^{-1}(3 x)=2 \pi \] The solution set is $\square$ 3. 6 (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed.)

Solution

Solution Steps

To solve the equation $3 \cos^{-1}(3x) = 2\pi$, we first isolate $\cos^{-1}(3x)$ by dividing both sides by 3. Then, we find the cosine of both sides to solve for $3x$. Finally, we solve for $x$ by dividing by 3.

Step 1: Isolate $\cos^{-1}(3x)$

To solve the equation $3 \cos^{-1}(3x) = 2\pi$, we first isolate $\cos^{-1}(3x)$ by dividing both sides by 3: \[ \cos^{-1}(3x) = \frac{2\pi}{3} \approx 2.0944 \]

Step 2: Find the Cosine Value

Next, we find the cosine of both sides to solve for $3x$: \[ 3x = \cos\left(\frac{2\pi}{3}\right) \approx -0.5000 \]

Step 3: Solve for $x$

Finally, we solve for $x$ by dividing both sides by 3: \[ x = \frac{-0.5000}{3} \approx -0.1667 \]