Problem 21 - Calculus of a Single Variable

Question

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\sin \left(\arctan \frac{3}{4}\right)\) (b) \(\sec \left(\arcsin \frac{4}{5}\right)\)

Step-by-Step Solution

Answer

(a) \(\sin \left(\arctan \frac{3}{4}\right) = \frac{3}{5} \) and (b) \(\sec (\arcsin \frac{4}{5})= \frac{5}{3} \)

Step 1: Evaluate \(\sin \left(\arctan \frac{3}{4}\right)\)

Given \(\arctan \frac{3}{4}\) we know opposite side length is 3 and adjacent side length is 4 in a right triangle. Due to Pythagorean theorem, the hypotenuse can be calculated as: \( \sqrt{3^2 + 4^2} = 5 \). Consequently, \( \sin \theta = \frac{opposite}{hypotenuse} = \frac{3}{5} \). So, \(\sin \left(\arctan \frac{3}{4}\right)=\frac{3}{5} \).

Step 2: Evaluate \(\sec (\arcsin \frac{4}{5})\)

Here, the argument of the trigonometric function is \(\arcsin \frac{4}{5}\). In a right triangle, this implies the hypotenuse is 5 and the side opposite of the angle is 4. Using the Pythagorean theorem again, the remaining side adjacent to the angle is \( \sqrt{5^2-4^2} = 3 \). Consequently, the secant is the reciprocal of the cosine function, or \( \frac{hypotenuse}{adjacent} = \frac{5}{3} \). Hence, \(\sec (\arcsin \frac{4}{5})= \frac{5}{3} \).