Trigonometry Examples

$\sec (\arctan (3 x))$

Step 1

Draw a triangle in the plane with vertices $(1, 3 x)$ , $(1, 0)$ , and the origin. Then $\arctan (3 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 3 x)$ . Therefore, $\sec (\arctan (3 x))$ is $\sqrt{1 + {(3 x)}^{2}}$ .

$\sqrt{1 + {(3 x)}^{2}}$

Step 2

Apply the product rule to $3 x$ .

$\sqrt{1 + 3^{2} x^{2}}$

Step 3

Raise $3$ to the power of $2$ .

$\sqrt{1 + 9 x^{2}}$