Problem 55 - Precalculus: A Concise Course

Question

Find the exact value of the expression. (Hint: Sketch a right triangle.) $$ \sin \left(\arctan \frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression \(\sin(\arctan\frac{3}{4})\) is \(\frac{3}{5}\)

Step 1: Interpret the arctan function and build a right triangle

An angle whose tangent is 3/4 can be represented by a right triangle where the side opposite the angle is of length 3 and the adjacent side is of length 4. Draw this triangle.

Step 2: Determine the length of the hypotenuse

With the Pythagorean theorem, calculate the length of the hypotenuse \(h\). Using the theorem \(a^2 + b^2 = h^2\), where \(a\) and \(b\) are the lengths of the adjacent and opposite sides respectively (3 and 4 in our case). The length will be sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.

Step 3: Calculate the sine of the angle

The sine of an angle in a right triangle is given by the ratio of the length of the side opposite the angle to the length of the hypotenuse. Here, the sine of our angle is 3/5.

Key Concepts

Right Triangle

When solving mathematical problems involving trigonometric functions, the right triangle plays a crucial role. A right triangle is one where one of the angles is exactly 90 degrees. This type of triangle has unique properties that make it very useful for calculations involving trigonometry.
In the context of the exercise, the right triangle helps us visualize and find the relationships between the sides and angles. For this particular problem, where we are dealing with an angle obtained from the arctangent of a fraction, we can start by sketching a right triangle.
In our triangle, the side opposite to the angle has a length of 3, and the adjacent side has a length of 4. The hypotenuse, which is the side opposite the right angle, can be calculated using the Pythagorean theorem. This practical approach allows us to easily determine various trigonometric ratios, like sine, without engaging in complex calculations.

• The side "opposite" is perpendicular to the adjacent side.
• The "adjacent" side lies next to the angle in question.
• Using the foundational Pythagorean theorem, we can calculate the hypotenuse: \[ h = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 .\]

Arctangent

Arctangent, also known as inverse tangent, is a trigonometric function that allows us to determine an angle given the ratio of the opposite side to the adjacent side in a right triangle. It is often denoted as \( \arctan \) or \( \tan^{-1} \).
In this problem, the angle whose tangent is given by the ratio \( \frac{3}{4} \) is determined using arctangent. This means that if the tangent of the angle equals \( \frac{3}{4} \), then the angle itself is \( \arctan \frac{3}{4} \).
Creating a right triangle based on this ratio provides a visual representation of the problem, aiding in calculating other trigonometric functions such as sine or cosine.

• The angle we find using \( \arctan \) is the angle formed by the ratio of the opposite (3) to the adjacent (4).
• This angle is crucial for determining further trigonometric functions related to our exercise.

Sine

The sine function in trigonometry gives the ratio of the length of the side of a right triangle opposite a given angle to its hypotenuse. This function is fundamental because it relates the angle to the sides of the triangle.
In our problem, we are required to find \( \sin \left( \arctan \frac{3}{4} \right) \). We've determined the structure of the right triangle using arctangent and calculated the hypotenuse to be 5 through the Pythagorean theorem.
The sine of the angle is then calculated by taking the length of the opposite side (which is 3) and dividing it by the hypotenuse (which is 5). So, \( \sin \left( \arctan \frac{3}{4} \right) = \frac{3}{5} \). This simplifies the trigonometric evaluation using the given information efficiently.

• Sine is used frequently to transition between angle and ratio information in trigonometric problems.
• Understanding the relationship between the sides of the triangle is key for using sine.