Chapter 5, Problem 20E

Explanation of Solution

Tree of calls:

• Representing the calls of recursive function in a tree format is called as tree of calls.
• Values in each recursive call can be represented as children of parent node.
• For example, in first recursive call “$\sin \left(\frac{x}{3}\right)$, $\tan \left(\frac{x}{3}\right)$, and $\sin \left(\frac{x}{3}\right)$ are children of node $\sin x$.

Formulas for Trigonometric functions sine, cosine, and tangent:

Mathematical formula for calculating $\sin x$ is:

$\text{sin(x)}\text{=}\text{sin}\left(\frac{\text{x}}{\text{3}}\right)\frac{\left({\text{3-tan}}^{\text{2}}\left(\frac{\text{x}}{\text{3}}\right)\right)}{\left({\text{1+tan}}^{\text{2}}\left(\frac{\text{x}}{\text{3}}\right)\right)}$

Mathematical formula for calculating $\cos x$ is:

$\cos (x)=1-\sin \left(\frac{x}{2}\right)$

Mathematical formula for calculating $\tan x$ is:

$\text{tan(x)}\text{=}\frac{\text{sin(x)}}{\text{cos(x)}}$

From above formulas,

• $\sin x$ is derived into $\sin \left(\frac{x}{3}\right)$, $\tan \left(\frac{x}{3}\right)$ and $\tan \left(\frac{x}{3}\right)$
• $\cos x$ is derived into $\sin \left(\frac{x}{2}\right)$
• $\tan x$ is derived into $\sin x$ and $\cos x$

Tree of calls for $\sin x$.

Tree of calls for $\sin x$ till angle is less than or equal to $\frac{x}{18}$ is shown below.

Explanation:

• $\sin \left(x\right)$ is derived into $\sin \left(\frac{x}{3}\right)$, $\tan \left(\frac{x}{3}\right)$ and $\tan \left(\frac{x}{3}\right)$.
• $\tan \left(\frac{x}{3}\right)$ is derived into $\sin \left(\frac{x}{3}\right)$ and $\cos \left(\frac{x}{3}\right)$