Problem 12 - Algebra and Trigonometry

Question

\(1-12\) : Use an addition or subtraction formula to find the exact value of the expression, as demonstrated in Example \(1 .\) $$ \tan \frac{7 \pi}{12} $$

Step-by-Step Solution

Answer

The exact value of \(\tan \frac{7\pi}{12}\) is \(-2 - \sqrt{3}\).

Step 1: Identify Addition or Subtraction Formulas for Tangent

The tangent of a sum or a difference can be found using these formulas: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\) and \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\). We need to choose values of \(a\) and \(b\) for which the tangent values are known.

Step 2: Express \(\frac{7\pi}{12}\) as a Sum or Difference

\(\frac{7\pi}{12}\) can be rewritten using known angles: \(\frac{7\pi}{12} = \frac{3\pi}{12} + \frac{\pi}{4} = \frac{\pi}{4} + \frac{\pi}{3}\). Here, \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) are angles with known tangent values.

Step 3: Calculate Tangent of Known Angles

We know that \(\tan \frac{\pi}{4} = 1\) and \(\tan \frac{\pi}{3} = \sqrt{3}\). These will be used in the addition formula.

Step 4: Apply Tangent Addition Formula

Using the formula for the tangent of a sum: \(\tan(\frac{\pi}{4} + \frac{\pi}{3}) = \frac{\tan \frac{\pi}{4} + \tan \frac{\pi}{3}}{1 - \tan \frac{\pi}{4} \tan \frac{\pi}{3}}\). Substituting the known values: \(\frac{1 + \sqrt{3}}{1 - 1 \times \sqrt{3}} = \frac{1 + \sqrt{3}}{1 - \sqrt{3}}\).

Step 5: Simplify Expression with Rationalization

To simplify \(\frac{1 + \sqrt{3}}{1 - \sqrt{3}}\), multiply both numerator and denominator by the conjugate \(1 + \sqrt{3}\): \(\frac{(1+\sqrt{3})^2}{(1-\sqrt{3})(1+\sqrt{3})}\).

Step 6: Perform and Simplify Calculations

Calculating: \((1+\sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}\). The denominator \((1-\sqrt{3})(1+\sqrt{3}) = 1 - 3 = -2\). So we have \(\frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}\).

Key Concepts

Tangent Addition Formula

The tangent addition formula is a key tool in trigonometry that allows us to find the tangent of the sum of two angles. This formula is expressed as:\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]It's important to use known angles to successfully apply this formula. By adding or subtracting angles that have exact tangent values, like \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\), you can simplify the problem significantly. These angles are convenient because their tangent values are easy to remember and widely used in trigonometric problems.
- For \(\tan \frac{\pi}{4} = 1\)- For \(\tan \frac{\pi}{3} = \sqrt{3}\)Using this formula, complex trigonometric expressions can be broken down into simpler parts, making it easier to work through the calculation and arrive at an exact solution.

Exact Trigonometric Values

Exact trigonometric values are special values of trigonometric functions that you should memorize. These often correspond to special angles like \(\frac{\pi}{6}, \frac{\pi}{4},\) and \(\frac{\pi}{3}\). Knowing these values off-hand is incredibly useful for solving problems quickly.For example:

• \(\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}\)
• \(\tan \frac{\pi}{4} = 1\)
• \(\tan \frac{\pi}{3} = \sqrt{3}\)

Learning the exact values for sine, cosine, and tangent at these angles allows you to plug these into formulas like the tangent addition formula with confidence. These values create a "toolbox" you can rely on to simplify trigonometric expressions without a calculator.

Angle Sum Identities

Angle sum identities are formulas that provide a means of extending trigonometric evaluations to sums and differences of angles, not just individual ones. These identities include formulas for sine, cosine, and tangent.For the tangent function, this identity is expressed as:\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]Similarly, there are identities for sine and cosine:

• Sine: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
• Cosine: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)

These identities are crucial because they let you explore and compute the trigonometric functions for various composite angles. That means, instead of trying to find complex angles directly, you can break them down into simpler known angles and apply these identities. So, when faced with an angle like \(\frac{7\pi}{12}\), leveraging these identities can guide you to an exact result effectively. Understanding them means you can solve more intricate trigonometric problems effortlessly by mapping them back to basic principles.