Problem 64 - Precalculus Essentials

Question

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$\sin \left(\tan ^{-1} x\right)$$

Step-by-Step Solution

Answer

The expression \(\sin (\tan ^{-1} x)\) in terms of algebraic expression is \(x / \sqrt{1^2 + x^2}\).

Step 1: Understand the function

Here, we're dealing with \(\sin (\tan ^{-1} x)\). The inner function, \(\tan ^{-1} x\), refers to the arctangent of x, which is the angle whose tangent is x. Because the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, this tells us that we can form a right triangle where the length of the opposite side is x and the length of the adjacent side is 1.

Step 2: Apply Pythagorean theorem

Since we know the lengths of the opposite and adjacent sides, we can use the Pythagorean theorem to find the length of the hypotenuse. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This gives us \(\sqrt{1^2 + x^2}\).

Step 3: Determine the sine

The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. Given that we know the opposite side to be x and the hypotenuse to be \(\sqrt{1^2 + x^2}\), this gives us the ratio \(x / \sqrt{1^2 + x^2}\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one of its angles measuring exactly 90 degrees, known as a right angle. Understanding right triangles is crucial when learning about trigonometry, as trigonometric functions are often derived from their relationships within these triangles. In a right triangle, the three sides have specific names: the side opposite the right angle is the hypotenuse, and the other two sides are called the adjacent and opposite sides with respect to a given angle. These sides and angles play a key role in defining trigonometric functions, such as sine, cosine, and tangent.

For inverse trigonometric functions, remember that when we find the angle using functions like \(\tan^{-1}x\), we are essentially determining the angle whose tangent gives the ratio \frac{x}{1}\. This is why understanding how to represent angles and sides in right triangles is vital to solving trigonometric equations.

Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in Euclidean geometry. It relates the lengths of the sides of a right triangle. Simply put, in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides.

This theorem is usually expressed with the formula: \[ c^2 = a^2 + b^2\]where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

In the context of converting \(\sin(\tan^{-1}x)\) into an algebraic expression, the Pythagorean theorem allows us to calculate the length of the hypotenuse, \(\sqrt{1^2 + x^2}\). Knowing this length is crucial because it helps us determine the sine of our angle.

So, whenever you are given an inverse trigonometric function and you need to solve for the sine, cosine, or tangent, using the Pythagorean theorem to compute the hypotenuse or unknown side is an essential step.

Trigonometric Ratios

Trigonometric ratios are fundamental in understanding the relationships between the angles and sides of a right triangle. These ratios \(\sin, \cos,\ \text{and}\ \tan\) describe the relationships in an angle of a right triangle:

• Sine (\(\sin\)): Ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos\)): Ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): Ratio of the length of the opposite side to the adjacent side.

Applying these ratios can help solve a wide range of problems involving right triangles. For instance, when converting \(\sin(\tan^{-1}x)\) to an algebraic expression, we identify the sides of the corresponding right triangle first.

Knowing \(\tan^{-1}x\) implies that the ratio \(\frac{x}{1}\) relates to the opposite and adjacent sides. As a result, we derive the hypotenuse using the Pythagorean theorem. Finally, the sine of the angle is calculated using the trigonometric ratio for sine, resulting in \(\frac{x}{\sqrt{1^2 + x^2}}\).

Understanding and applying trigonometric ratios is crucial for anyone working with right triangles and inverse trigonometric functions.