Final answer:
To find the value of h(20) = tan(20), we can use the trigonometric ratios from the given figure. By applying the tangent ratio, we can establish the equation tan 20 = 9/a. Then, using trigonometric identities, we can simplify the expression and substitute the values of sin 20 and cos 20 from the figure to find the value of h(20).
Step-by-step explanation:
To evaluate the function h(x) = tan x at x = 20, we can use the trigonometric ratios from the given figure. Since h(x) = tan x, we need to find the value of tan 20. Looking at the figure, we can see that the opposite side (y) has a length of 9 and the adjacent side (x) has a length of a. Using the tangent ratio, tan 20 = y/x. Substituting the given values, tan 20 = 9/a.
To simplify this expression, we can use the fact that tan is the ratio of sin and cos. Using the identity sin²θ + cos²θ = 1, we can rewrite tan 20 = sin 20 / cos 20. From the given information, we have sin 20 = 2 sin 20 cos 20 and cos 20 = 1 - 2 sin² 20. Substituting these values, tan 20 = 2 sin 20 cos 20 / (1 - 2 sin² 20).
Now we can substitute the value of sin 20 in terms of tan 20. We have sin 20 = sin (atan 20) = (2 tan 20) / (1 + tan² 20). Substituting this value back into the equation, we get tan 20 = 2 (2 tan 20) / (1 + tan² 20) cos 20 / (1 - 2 (2 tan 20 / (1 + tan² 20))²). Simplifying this expression further, we get tan 20 = 4 tan 20 / (1 + tan² 20) cos 20 / (1 - 4 tan² 20 / (1 + tan² 20))².
Now we can substitute tan 20 back into the equation to get the final value of h(20). Using the given figure, we can determine the values of sin 20 and cos 20, and substitute them in the expression. Finally, we can simplify the expression to get the value of h(20) as a simplified fraction or decimal.