Questions: Use the figure to evaluate the function given that g(x) = cos x. g(2 alpha) g(2 alpha) = (Simplify your answer.)

Transcript text: Use the figure to evaluate the function given that $\mathrm{g}(\mathrm{x})=\cos \mathrm{x}$. \[ g(2 \alpha) \] $\mathrm{g}(2 \alpha)=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Find the value of cos(α)

The point on the unit circle is given as (-2/3, b). Since it lies on the unit circle x² + y² = 1, we have (-2/3)² + b² = 1. This implies b² = 1 - 4/9 = 5/9. Since b is the y-coordinate in the third quadrant, it must be negative. Therefore, b = -√5/3.

The cosine of the angle α is given by the x-coordinate of the point on the unit circle. So, cos(α) = -2/3.

Step 2: Find the value of cos(2α)

We need to find g(2α) which is equal to cos(2α). Using the double angle formula for cosine, we have cos(2α) = 2cos²(α) - 1.

Step 3: Substitute the value of cos(α)

Substituting cos(α) = -2/3, we get:

cos(2α) = 2(-2/3)² - 1 = 2(4/9) - 1 = 8/9 - 1 = (8 - 9)/9 = -1/9