Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\tan \left(\arccos \frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{7}}{3} \).

Step 1: Understand the Problem

We need to find the value of \( \tan \left(\arccos \frac{3}{4}\right) \). This expression involves using trigonometric identities to determine the tangent of an angle when given the cosine.

Step 2: Represent the Angle

Let \( \theta = \arccos \frac{3}{4} \). This means that \( \cos \theta = \frac{3}{4} \). We need to find \( \tan \theta \).

Step 3: Use Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \cos \theta = \frac{3}{4} \) into this identity:\[\sin^2 \theta + \left(\frac{3}{4}\right)^2 = 1\]

Step 4: Solve for \( \sin \theta \)

Simplify the equation from Step 3:\[\sin^2 \theta + \frac{9}{16} = 1\]Subtract \( \frac{9}{16} \) from both sides to find \( \sin^2 \theta \):\[\sin^2 \theta = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}\]Take the square root to find \( \sin \theta \):\[\sin \theta = \frac{\sqrt{7}}{4}\]

Step 5: Find \( \tan \theta \)

The tangent of an angle is the ratio of the sine and cosine:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}}\]Simplify the fraction:\[\tan \theta = \frac{\sqrt{7}}{3}\]

Step 6: Write the Final Answer

We have found that \( \tan \left(\arccos \frac{3}{4}\right) = \frac{\sqrt{7}}{3} \). This is the exact real number value of the given expression.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the fundamental trigonometric functions. It provides the ratio of the opposite side to the adjacent side in a right-angled triangle. This makes it very useful in many calculations involving angles and lengths.

• In terms of sine and cosine, the tangent function can be represented as: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
• Unlike sine and cosine, the tangent function has a period of \( \pi \) radians, meaning it repeats every \( \pi \) units along the x-axis.
• The graph of the tangent function exhibits vertical asymptotes where the cosine function is zero, making it undefined at those points.

By understanding the tangent function, you can analyze and solve problems involving angle measures and their respective trigonometric ratios. Calculations frequently use this function in conjunction with both sine and cosine values to fully describe geometric and physical phenomena.

Inverse Trigonometric Functions

Inverse trigonometric functions are the opposites of the regular trigonometric functions, helping us to find the angle when we know the function value. In the given exercise, the inverse cosine, represented as \( \arccos x \), is used to determine an angle whose cosine is \( x \).

• The range of \( \arccos x \) is \([0, \pi]\), providing angles in the first and second quadrants of the unit circle.
• The primary function of inverses is to "undo" the effect of the original trigonometric function and help retrieve the angle measure when its trigonometric value is known.
• In the context of our problem: \( \theta = \arccos \frac{3}{4} \) means we are looking for the angle \( \theta \) whose cosine value is exactly \( \frac{3}{4} \).

This concept allows us to handle problems where we initially know the trigonometric ratio, helping further to derive other related trigonometric values by using identities.

Pythagorean Identity

The Pythagorean Identity is a crucial component in trigonometry and is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This fundamental relation stems from the Pythagorean Theorem applied to a unit circle. Here are some key points about this identity:

• This identity allows us to solve for one trigonometric function if the other is known. For instance, if \( \cos \theta \) is given, you can rearrange the identity to solve for \( \sin \theta \) and vice versa.
• It reinforces the idea that the sum of the squares of the sine and cosine of any angle will always equal one, showing a consitant relationship between these functions.
• In practice: When given \( \cos \theta = \frac{3}{4} \), you can find \( \sin \theta \) using \( \sin^2 \theta = 1 - \cos^2 \theta \).
• After determining \( \sin \theta \), use it to find other trigonometric values, such as \( \tan \theta \), through the relation \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Understanding and applying the Pythagorean Identity is essential for evaluating trigonometric expressions and ensuring accurate computations in mathematical problems.