Problem 7 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$\tan 15^{\circ}$$

Step-by-Step Solution

Answer

\( \tan 15^{\circ} = 2 - \sqrt{3} \).

Step 1: Identify the Formula

To find the value of \( \tan 15^{\circ} \), we use the tangent subtraction formula. The general formula for tangent subtraction is given by: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \].

Step 2: Choose Appropriate Angles

We can write \( 15^{\circ} \) as the difference of two angles with known tangent values. Let's use \( 45^{\circ} - 30^{\circ} = 15^{\circ} \). Therefore, we'll set \( a = 45^{\circ} \) and \( b = 30^{\circ} \).

Step 3: Apply the Tangent Subtraction Formula

Substitute \( a = 45^{\circ} \) and \( b = 30^{\circ} \) into the tangent subtraction formula. We have \[ \tan 15^{\circ} = \frac{\tan 45^{\circ} - \tan 30^{\circ}}{1 + \tan 45^{\circ} \tan 30^{\circ}} \].

Step 4: Use Known Tangent Values

Recall that \( \tan 45^{\circ} = 1 \) and \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \). Substitute these values into the equation: \[ \tan 15^{\circ} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} \].

Step 5: Simplify the Expression

Simplify the numerator and denominator:- Numerator: \( 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \).- Denominator: \( 1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}} \).Thus, \[ \tan 15^{\circ} = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \].

Step 6: Simplify by Multiplying by Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the square root:\[ \tan 15^{\circ} = \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{(\sqrt{3} - 1)^2}{2} \].

Step 7: Final Calculation

Calculate \( (\sqrt{3} - 1)^2 \):\[ \tan 15^{\circ} = \frac{(\sqrt{3})^2 - 2\cdot\sqrt{3}\cdot1 + 1^2}{2} = \frac{3 - 2\sqrt{3} + 1}{2} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \].

Key Concepts

Tangent Subtraction Formula

The tangent subtraction formula is incredibly useful when trying to find the tangent of an angle that is a difference of two other angles.
This formula can be expressed as:

• \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]

It allows us to break down a complex problem into simpler parts by leveraging angles with known values. In this scenario, we're finding \( \tan 15^{\circ} \) by using angles \( 45^{\circ} \) and \( 30^{\circ} \).
Remember to substitute correctly and perform each operation step by step for accuracy.

Angle Conversion

Angle conversion plays a crucial role in the study of trigonometry. Knowing how to express an angle as a sum or difference of standard angles helps in applying formulas.
For the exercise, we converted \( 15^{\circ} \) into \( 45^{\circ} - 30^{\circ} \). This step is essential as we need angles with known trigonometric values for further calculations. It showcases the importance of recognizing angles as different combinations to simplify our math problems.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions are foundational in trigonometry.
The specific values used in this solution include:

• \( \tan 45^{\circ} = 1 \)
• \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \)

These values help to make complex calculations more manageable.
Rather than using a calculator for approximate values, exact values ensure precision and simplicity in problem-solving.

Simplifying Square Roots

Simplifying square roots can often seem daunting, but it's a crucial skill in trigonometry.
During the exercise, square roots are simplified to eliminate them from the denominator, ensuring a cleaner final expression. Multiplying by the conjugate is a common technique to achieve this.

• Conjugate: Given \( a + b\sqrt{c} \), its conjugate is \( a - b\sqrt{c} \).

For \( \tan 15^{\circ} \), we multiplied by the conjugate of the denominator, \( \sqrt{3} + 1 \), leading to a rationalized and simplified result: \( 2 - \sqrt{3} \).