Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of each expression, if possible, without using a calculator. (a) \(\arccos (-\sqrt{3})\) (b) \(\arcsin \frac{\sqrt{2}}{2}\)

Step-by-Step Solution

Answer

\(\arccos (-\sqrt{3}/2) = 5π/6\) and \(\arcsin (\sqrt{2}/2) = π/4\)

Step 1: Analyze the arccosine function

Arccos, or inverse cosine, gives us an angle whose cosine is the input value. The range for arccos is \(0 ≤ θ ≤ π\). Here, the input value is \(-\sqrt{3}/2\). The cosine of \(π/6\) is \(\sqrt{3}/2\) and of \(5π/6\) is \(-\sqrt{3}/2\). So, \(\arccos (-\sqrt{3}/2)\) will take the value of \(5π/6\).

Step 2: Analyze the arcsine function

Arcsin, or inverse sine, gives us an angle whose sine is the input value. The range for arcsin is \(-π/2 ≤ θ ≤ π/2\). Here, the input value is \(\sqrt{2}/2\). The sine of both \(π/4\) and \(-π/4\) are \(\sqrt{2}/2\). However, since \(-π/4\) is not in the domain of arcsin, the only possibility left is \(π/4\). So, \(\arcsin (\sqrt{2}/2)\) will take the value of \(π/4\).

Key Concepts

Arccosine Function

The arccosine function, often represented as \( \arccos \), is an inverse trigonometric function. It helps you find an angle from a given cosine value. If you have a cosine value, you can use \( \arccos \) to figure out which angle produced that cosine.

One thing to remember about \( \arccos \) is its range. It always results in an angle between 0 and \( \pi \) radians, which is between 0 and 180 degrees. This means, no matter the input, the output (the angle) will fall within this range. It helps provide a unique angle for any given cosine value within this range.

• If the input is positive or zero, \( \arccos \) gives an angle on the upper half of the unit circle, meaning between 0 and 90 degrees.
• If the input is negative, the angle is on the lower half, between 90 and 180 degrees.

Using this information, you can determine that for \( \arccos (-\sqrt{3}/2) \), the angle is \( 5\pi/6 \) because the cosine of this angle within \( 0 \) to \( \pi \) is \( -\sqrt{3}/2 \).

Arcsine Function

The arcsine function, written as \( \arcsin \), is another key inverse trigonometric function. It allows you to determine an angle when you already know the sine of that angle. Understanding \( \arcsin \) is crucial for working with trigonometric values in reverse.

The range of the \( \arcsin \) function is between \(-\pi/2\) and \(\pi/2\), which translates to angles from -90 to 90 degrees. This characteristic ensures that there is a one-to-one correspondence between sine values and angles.

• For positive sine values, \( \arcsin \) gives an angle in the first quadrant (0 to 90 degrees).
• For negative sine values, it produces angles in the fourth quadrant (-90 to 0 degrees).

In our case, \( \arcsin(\sqrt{2}/2) \) results in \( \pi/4 \), as the sine of \( \pi/4 \) is \( \sqrt{2}/2 \) and \( \pi/4 \) is within the \( \arcsin \) range.

Trigonometric Values

Trigonometric values relate to the sine and cosine of angles in a right-angled triangle or the unit circle. Essential understanding of these values can help you solve equations like those with the \( \arccos \) and \( \arcsin \) functions.

Using the unit circle, the most commonly referenced angles are \( 0, \pi/6, \pi/4, \pi/3, \pi/2, \) and their respective angles in radians such as \( \pi, \) and \( 2\pi \). Each of these angles has known sine and cosine values that are crucial when working with inverse trigonometric functions.

• For \( \pi/6,\) the cosine is \( \sqrt{3}/2 \) and sine is \( 1/2\).
• For \( \pi/4, \) both sine and cosine are \( \sqrt{2}/2 \).
• For \( \pi/3, \) the sine is \( \sqrt{3}/2 \) and cosine is \( 1/2 \).

When solving for inverse functions like \( \arccos \) and \( \arcsin \), knowing these trigonometric values helps directly match angles to given trigonometric results. This knowledge allows you to work efficiently without always needing a calculator, especially within concepts revolving around inverse trigonometric functions.