Problem 22 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \tan \left(\frac{\pi}{3}+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression \(\tan \left(\frac{\pi}{3}+\frac{\pi}{4}\right)\) is \(-1 -\sqrt{3}\).

Step 1: Identifying the Identity

We start by understanding the formula for the sum of angles of tangents, which is given by \(\tan(A+B)= \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}\). Here, we have \(A = \frac{\pi}{3}\) and \(B = \frac{\pi}{4}\).

Step 2: Apply the Identity

We now substitute the values of \(A\) and \(B\) into the identity. Which gives us \(\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) = \frac{\tan (\frac{\pi}{3}) + \tan (\frac{\pi}{4})}{1 - \tan (\frac{\pi}{3}) \cdot \tan (\frac{\pi}{4})}\).

Step 3: Substitute values from the Unit Circle

We know that \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(\frac{\pi}{4}) = 1\). Hence we substitute these values into the expression which grants us \(\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}\).

Step 4: Simplify the Expression

Further simplifying this we get \(\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}\).

Step 5: Rationalize the Denominator

To get the final answer, we rationalize the denominator. After rationalizing, we get \(\tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) = -1 - \sqrt{3}\).

Key Concepts

Sum and Difference Formulas

The sum and difference identities are fundamental tools in trigonometry. These formulas help in finding the trigonometric function values of angles that are either sums or differences of known angles.
To compute \( \tan(A + B) \), the sum formula for tangent is utilized:

• \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \]

This identity simplifies the process of working with angles that are not commonly memorized or directly found on the unit circle. Understanding how to derive these identities and apply them effectively in problems can enormously simplify complex trigonometric calculations.

Tangent Function

The tangent function, often abbreviated as \( \tan \), is one of the six primary trigonometric functions. It is defined in terms of sine and cosine:

• \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

The tangent of an angle in the unit circle is the slope of the line from the origin to the point corresponding to the angle on the unit circle.
Another view is that \( \tan \theta \) represents the ratio of the opposite side to the adjacent side in a right-angled triangle. For angles like \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \), the tangent values are well-known:

• \( \tan \left(\frac{\pi}{3}\right) = \sqrt{3} \)
• \( \tan \left(\frac{\pi}{4}\right) = 1 \)

These known values allow us to calculate the tangent of combined angles like \( \left(\frac{\pi}{3} + \frac{\pi}{4}\right) \).

Unit Circle

The unit circle is a powerful visual tool in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate system. The unit circle helps to visualize the coordinates of angles and to find trigonometric values for common angles.
In the case of our problem, the unit circle aids in determining the tangent values of \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \).

• For \( \frac{\pi}{3} \): the coordinates are \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \)
• For \( \frac{\pi}{4} \): the coordinates are \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \)

Using the x- and y-coordinates, you can find the tangent by dividing the y-coordinate by the x-coordinate, simplifying the computation of trigonometric expressions with angles not immediately given on the circle.

Rationalizing Denominator

Rationalizing the denominator is a common mathematical technique used to eliminate irrational numbers from the denominator of a fraction.
This simplifies calculations and expressions, particularly in trigonometry.
When solving \( \tan\left(\frac{\pi}{3}+\frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \), we end with a simplified solution by rationalizing:

• Multiply the numerator and the denominator by the conjugate of the denominator \( 1 + \sqrt{3} \).

This step results in a rational denominator, giving us the final simplified expression \(-1 - \sqrt{3} \). Understanding this process aids in clearing complex fractions into more manageable forms.