Consider the circle whose center is (0,0) and whose radius is r =3. Consider a point (a,b) on this circle, with central angle θ =π/6 . Draw the triangle with vertices (3,0), (0,0), and (a,b). Find the exact values for a and b. Explain how you found these values.

Did you draw the figure? It should be clear that

a = 3cosθ
b = 3sinθ

Did you know?

Did you know that for a circle with center (0,0) and radius r = 3, if we consider a point (a,b) on the circle with a central angle θ = π/6, we can find the exact values for a and b? To find these values, we draw a triangle with vertices (3,0), (0,0), and (a,b). The side connecting (3,0) and (0,0) is the x-axis, which means it has a length of 3 units. Since the central angle is π/6, we can divide the x-axis length (3) by 2 to get the adjacent side length of the angle. This gives us a = 3/2. To find the length of the opposite side (b), we can use the formula sine(theta) = opposite/hypotenuse. In this case, sine(π/6) = b/3. By rearranging the equation, we find that b = 3 * sine(π/6) = 3 * 1/2 = 3/2. Therefore, the exact values for a and b in the triangle are a = 3/2 and b = 3/2. This demonstrates how we can use trigonometry and geometry to find precise values on a circle.