91Ó°ÊÓ

Inverse trigonometric functions allow us to find angles when given a trigonometric ratio. They are the reverse operations of regular trigonometric functions. For instance,

• the inverse of the sine function is denoted as \ \(\sin^{-1}(x)\ \) and is often called arcsin,
• while the inverse of the cosine function is \ \(\cos^{-1}(x)\ \) (also known as arccos),
• and similarly, \ \(\tan^{-1}(x)\ \) or arctan is the inverse of the tangent function.

These functions are crucial in determining angles, especially in geometric and trigonometric calculations.
In the original exercise, \ \(\tan^{-1}(x)\ \) indicates the angle whose tangent is \ \(x\ \). This angle helps us easily relate the tangent ratio back to a specific angle's measure in a right triangle.

Right triangles are fundamental in trigonometry and geometry. A right triangle is defined as a triangle with one 90-degree angle. The remaining two angles are acute and must sum up to 90 degrees.
In a right triangle:

• the longest side is called the hypotenuse,
• one side is called the opposite (the side opposite the angle of interest),
• and the other is the adjacent side (next to the angle of interest).

The trigonometric functions sine, cosine, and tangent are defined based on the ratios of these sides.
When you encounter expressions like \ \(\tan^{-1}(x)\ \), it's your cue to visualize or draw a right triangle where tangent (opposite over adjacent) equals \ \(x\ \). This exercise uses a triangle where one side adjacent to the angle is set to 1, and the opposite side is \ \(x\ \).

The Pythagorean theorem is a cornerstone principle in geometry that applies to right triangles. Named after the ancient Greek mathematician Pythagoras, the theorem states: In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This can be expressed as:

• \ \(a^2 + b^2 = c^2\ \), where \ \(c\ \) represents the hypotenuse,
• and \ \(a\ \) and \ \(b\ \) are the other two sides.

In our exercise, after identifying the sides of the right triangle as \ \(x\ \) and 1, we apply the Pythagorean theorem to find the hypotenuse: \ \(\sqrt{x^2 + 1}\ \).
With this, we can determine other trigonometric ratios like cosine using these side lengths, such as \ \(\cos(\tan^{-1}(x)) = \frac{1}{\sqrt{x^2 + 1}}\ \), as derived in the exercise.