Problem 11 - Algebra and Trigonometry

Question

Find the exact value of each expression. $$ \cos ^{-1} 0 $$

Step-by-Step Solution

Answer

\(\cos^{-1} 0 = \frac{\pi}{2}\)

Step 1: Understanding the inverse cosine function

The cosine function gives the ratio of the adjacent side to the hypotenuse in a right angled triangle. It is defined as \( \cos \theta = \frac{adjacent}{hypotenuse} \). Given an angle, it will return the cosine of that angle. But the inverse cosine function, also known as the arccosine, is the opposite of the cosine function. It takes a number as input and returns the angle whose cosine is equal to the input number. In other words, if \( y = \cos x \) then \( x = \cos^{-1} y \). It effectively allows us to determine the angle from the cosine of the angle.

Step 2: Applying the inverse function

Given \( \cos^{-1} 0 \), we have to determine the angle whose cosine is 0. From the unit circle or by recalling the cosine function, we know that the cosine of the angle is 0 at two points - at \( \frac{\pi}{2} \) and at \( \frac{3\pi}{2} \). However, the range of \( \cos^{-1} x \) is \( [0, \pi] \), which means the angle we are looking for must fall within this range. So, although the cosine of \( \frac{3\pi}{2} \) is also 0, it falls outside the range of \( \cos^{-1} x \) and so is not considered here.

Step 3: Finding the exact value

From the previous step, we know that the cosine of \( \frac{\pi}{2} \) is 0. And since it falls within the range of \( \cos^{-1} x \), we can conclude that:\( \cos^{-1} 0 = \frac{\pi}{2} \).

Key Concepts

Understanding the Unit Circle

The unit circle is an essential tool in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. This circle helps us understand angles and their corresponding trigonometric values.
The unit circle allows one to easily find sine, cosine, and tangent values for different angles. The angle is measured from the positive x-axis, moving counter-clockwise.

• At 0 radians or 360 degrees, the point on the unit circle is (1, 0).
• At \( \frac{\pi}{2} \) or 90 degrees, it is (0, 1).
• At \( \pi \) or 180 degrees, it's (-1, 0).

These points directly relate to the cosine and sine values of the angles. For instance, the point (0, 1) at \( \frac{\pi}{2} \) shows us the cosine is 0 and the sine is 1.

Exploring Trigonometric Functions

Trigonometric functions are all about the relationships between the angles and sides of triangles. The main functions include sine, cosine, and tangent.

• Cosine (cos): This function compares the length of the adjacent side to the hypotenuse in a right angled triangle. Mathematically, it is expressed as \( \cos \theta = \frac{adjacent}{hypotenuse} \).
• Sine (sin): This one compares the opposite side to the hypotenuse, \( \sin \theta = \frac{opposite}{hypotenuse} \).
• Tangent (tan): It relates the opposite side to the adjacent side, \( \tan \theta = \frac{opposite}{adjacent} \).

These functions help us navigate angles and predict the behavior of waves, among other applications.

Decoding Arccosine (Inverse Cosine Function)

Arccosine is an inverse trigonometric function, often denoted as \( \cos^{-1} \) or arccos. It helps us find the angle when we know the cosine value.
If \( y = \cos x \), then \( x = \cos^{-1} y \).
This means arccos takes a value between -1 and 1 and returns an angle between 0 and \( \pi \) radians (or 0 to 180 degrees). Arccos is particularly helpful when you need to reverse-engineer from a cosine value back to the original angle.
In our example, \( \cos^{-1} 0 = \frac{\pi}{2} \) because the cosine of \( \frac{\pi}{2} \) is 0.

Right Angled Triangles in Trigonometry

Right angled triangles are the foundation of trigonometry. These triangles have one angle that's exactly 90 degrees.
Trigonometric functions are based on these triangles, specifically focusing on the relationships between the sides.

• The hypotenuse is the side opposite the right angle, and is always the longest side.
• The adjacent side is next to the angle of interest.
• The opposite side is across from the angle of interest.

Understanding these terms helps in calculating trigonometric functions for a given angle. By knowing two sides, you can calculate a function like cosine and find an unknown angle using inverse functions like arccos.