Problem 22 - Precalculus with Calculus Previews

Question

Use a sum or difference formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \tan \frac{17 \pi}{12} $$

Step-by-Step Solution

Answer

\( \tan \frac{17\pi}{12} = -2 - \sqrt{3} \).

Step 1: Break Down the Angle into Known Angles

The angle \( \frac{17\pi}{12} \) can be expressed as the sum of known angles \( \frac{4\pi}{3} + \frac{\pi}{4} \). Both of these angles have known tangent values, which we can utilize in the sum formula.

Step 2: Apply Tangent Sum Formula

The tangent sum formula is \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). Here, \( a = \frac{4\pi}{3} \) and \( b = \frac{\pi}{4} \). We need to find \( \tan \frac{4\pi}{3} \) and \( \tan \frac{\pi}{4} \).

Step 3: Calculate Individual Tangent Values

Calculate \( \tan \frac{4\pi}{3} \). The equivalent angle in degrees is 240°, and \( \tan 240° = \tan 60° = \sqrt{3} \). Since 240° is in the third quadrant, \( \tan 240° = \sqrt{3} \).Calculate \( \tan \frac{\pi}{4} \). This is a well-known angle: \( \tan \frac{\pi}{4} = 1 \).

Step 4: Substitute into the Formula

Substitute the values into the tangent sum formula:\[\tan \left(\frac{17\pi}{12}\right) = \tan \left(\frac{4\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1}\]

Step 5: Simplify the Expression

Simplify the expression:\[\tan \left(\frac{17\pi}{12}\right) = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}\]Multiply numerator and denominator by the conjugate of the denominator \((1 + \sqrt{3})\):\[\tan \left(\frac{17\pi}{12}\right) = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{(\sqrt{3} + 1 + 3 + \sqrt{3})}{1 - 3}\]\[= \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}\]

Step 6: Conclusion: Result

The exact value of \( \tan \frac{17\pi}{12} \) using the sum formula is \(-2 - \sqrt{3}\).

Key Concepts

Tangent Sum Formula

The tangent sum formula is a handy tool when dealing with angles that combine more than one component angle. This formula is written as:\[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\]By using this formula, we can find the tangent value for angles that are not directly linked to the unit circle, simply by decomposing them into known angles. For example, if you need to find \( \tan(\frac{17\pi}{12}) \), you can break it down to \( \tan(\frac{4\pi}{3} + \frac{\pi}{4}) \). This way, you can use known values for simpler angles, and

• Apply the formula
• Simplify the terms

This approach makes complex trigonometric calculations more manageable and avoids unnecessary complications in calculations. Let's explore more on how breaking down angles helps in mathematical calculations.

Angle Addition Formulas

Angle addition formulas provide a method to calculate the trigonometric functions of sums or differences of angles. They are essential because they allow us to find exact values without a calculator. The tangent addition formula featured here is part of this broader family of identities. For example, given an angle \( \frac{17\pi}{12} \), it isn't directly on the unit circle.By expressing it as:

• \( \frac{4\pi}{3} + \frac{\pi}{4} \)

you simplify the calculation by dealing with angles you already know. This can also be applied to sine and cosine functions using their respective addition formulas. By using addition formulas, you're efficiently handling composite angles by piecing them into simpler, known components, which in turn, leads to less computational and theoretical error.

Exact Trigonometric Values

Exact trigonometric values refer to well-known values for specific angles like \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{5\pi}{6} \), among others. It's necessary to be familiar with these values as they help in finding properties of other angles due to their consistent use in formulas.Consider this key information:

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

These values come in handy when we substitute them into equation formulas like the tangent sum formula.When tackling exercises, relying on exact values rather than approximations is crucial, as it maintains the precision of your calculations. For instance, when solving for \( \tan(\frac{17\pi}{12}) \) in our example, leveraging exact values simplifies and streamlines the whole solutionAlways re-familiarize yourself with the core trigonometric values, as this aids both understanding and solving trig functions effortlessly.