91Ó°ÊÓ

Inverse trigonometric functions are used to find angles when given a trigonometric ratio. For example, if we know that the cosine of an angle is \(\frac{\sqrt{3}}{2}\), we can use the inverse cosine function, written as \(\cos^{-1}\), to determine the angle itself. In this case, the angle is \(60\) degrees or \(\frac{\pi}{3}\) radians.
Inverse functions "undo" what the regular trigonometric functions do. They take a ratio and give an angle:

• \(\sin^{-1}\) gives the angle whose sine is a given number.
• \(\cos^{-1}\) provides the angle whose cosine is a particular value.
• \(\tan^{-1}\) returns the angle whose tangent is specified.

These functions are often crucial in solving right triangle problems, especially when specific angle measures are unknown, but certain sides or lengths are. In this exercise, determining \(\cos^{-1} \frac{\sqrt{3}}{2}\) allows us to find an exact angle to substitute into the next steps of the solution.

Right triangle trigonometry involves understanding the relationships between a triangle's angles and sides. In a right triangle, where one angle is \(90\) degrees, the remaining angles and side lengths follow specific trigonometric ratios. For instance, in our exercise, we sketch a right triangle with one angle as \(\frac{\pi}{3}\).
Given the original hint to use a right triangle, we recall that:

• The hypotenuse is across from the right angle.
• The opposite side is across the angle we're interested in.
• The adjacent side lies next to the angle but is not the hypotenuse.

By visualizing or sketching a right triangle, we can better understand these relationships:

• Cosine of an angle relates the adjacent side to the hypotenuse.
• Sine of an angle involves the opposite side and the hypotenuse.

The right triangle in our exercise helps clarify the cosine value of \(\frac{\sqrt{3}}{2}\). This ratio directly shows which side lengths correspond to the given angle in a standard right triangle.

The Pythagorean Theorem is a cornerstone of right triangle geometry. It relates the squares of the lengths of a triangle's sides: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. This theorem is especially useful when two side lengths are known, helping to find the third.
In our exercise:

• We're given an adjacent side, \(\sqrt{3}\), and the hypotenuse, \(2\).
• We apply the Pythagorean Theorem to find the opposite side: \(b = \sqrt{2^2 - (\sqrt{3})^2}\).
• The calculation reveals that the length of the opposite side is \(1\).

This ability to find a missing side length allows us to complete any necessary trigonometric calculations, like finding sine or cosecant. By fully understanding the Pythagorean Theorem, students can confidently tackle related problems and explore the links between geometry and trigonometry in triangles.