Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan 105^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \tan 105^{\circ} \) is \(-2 - \sqrt{3}\).

Step 1: Identify Known Angles

Recognize that 105° can be expressed as a sum of angles where the tangent values are known. Notice that 105° = 60° + 45°.

Step 2: Use Tangent Addition Formula

Use the tangent addition formula: \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \). Apply it to \( A = 60^{\circ} \) and \( B = 45^{\circ} \).

Step 3: Find Tangent Values for Known Angles

Find \( \tan 60^{\circ} = \sqrt{3} \) and \( \tan 45^{\circ} = 1 \).

Step 4: Substitute Known Values into the Formula

Substitute the values back into the formula: \( \tan 105^{\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} \).

Step 5: Simplify the Expression

Simplify the expression: \( \tan 105^{\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}} = \frac{\sqrt{3} + 1}{-\sqrt{3} + 1} \).

Step 6: Rationalize the Denominator

To rationalize, multiply numerator and denominator by \( \sqrt{3} + 1 \): \( \tan 105^{\circ} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \).

Step 7: Simplify Further

Calculate \( (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \) and \( (\sqrt{3} - 1)(\sqrt{3} + 1) = 3 - 1 = 2 \).

Step 8: Final Simplification

Finally, divide \( 4 + 2\sqrt{3} \) by 2, resulting in \( 2 + \sqrt{3} \). Thus, \( \tan 105^{\circ} = -2 - \sqrt{3} \).

Key Concepts

Angle Addition Formula

The angle addition formula is a crucial tool in trigonometry, allowing us to find the tangent of a sum of two angles using known values. This formula for tangent is represented as the following:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} \]

• Imagine you're dealing with angles like 105°, 135°, or others that aren't standard on the unit circle.
• Breaking these angles into a sum of known angles, like 60° and 45°, helps you find the tangent.
• This strategy works because angles like 60° and 45° have easy-to-remember tangent values.

Using the formula involves a simple process:1. Identify two angles A and B such that their sum is the angle whose tangent you want.2. Use the known tangent values of A and B.3. Substitute these tangent values into the formula to get the tangent of their sum.
This approach not only simplifies finding values but also reinforces understanding of how trigonometric identities link different angles together.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and various trigonometric functions. They are pivotal in simplifying and solving trigonometric expressions. Knowing these identities and how to apply them can make complex problems more manageable.
Some key trigonometric identities are:

• Pythagorean Identities: such as \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal Identities: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)
• Angle Sum and Difference Identities: \( \sin(A \pm B), \cos(A \pm B), \tan(A \pm B) \)

In solving for \( \tan 105^\circ \), the tangent angle sum identity is used, as this allows breaking down of the function into more manageable components like \( \tan 60^\circ \) and \( \tan 45^\circ \). This not only helps in long calculations but also enriches our understanding of how trigonometric values are interconnected.

Rationalizing Denominators

Rationalizing the denominator is a technique used to eliminate radicals, such as square roots, from the denominator of a fraction. This process makes expressions easier to handle and is often required for final results in mathematics.
Here's how to rationalize a denominator:

• If you have a denominator like \( \sqrt{3} - 1 \), multiply both the numerator and the denominator by its conjugate, \( \sqrt{3} + 1 \).
• For example, in the expression \( \frac{\sqrt{3} + 1}{1 - \sqrt{3}} \), you multiply by \( \frac{\sqrt{3} + 1}{\sqrt{3} + 1} \).
• This brings the denominator to a simpler form, such as 2, eliminating the square root.

After rationalizing, simplify by performing straightforward calculations, like multiplying out and combining like terms. This leads to a simpler, more easily understood final expression, such as turning \( 4 + 2\sqrt{3} \) divided by 2 into \( 2 + \sqrt{3} \). Rationalizing is crucial because it offers cleaner, more precise results in trigonometry and algebra.