Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{5 \pi}{12}$$

Step-by-Step Solution

Answer

The exact values are: \(sin(\frac{5 \pi}{12}) = \frac{\sqrt{2} + \sqrt{6}}{4}\), \(cos(\frac{5 \pi}{12}) = \frac{\sqrt{6} - \sqrt{2}}{4}\), \(tan(\frac{5 \pi}{12}) = - \sqrt{2} + \sqrt{6}\)

Step 1: Splitting the Angle

The angle \(\frac{5 \pi}{12}\) can be split into the sum of two angles, \(\frac{2 \pi}{12}\) + \(\frac{3 \pi}{12}\), which simplifies to \(\frac{\pi}{6}\) + \(\frac{\pi}{4}\), whose sine and cosine values are known.

Step 2: Finding the Sine Value

Use the sine formula for the sum of two angles, where \(sin(a+b) = sin(a)cos(b) + cos(a)sin(b)\). Here, \(a=\frac{\pi}{6}\) and \(b=\frac{\pi}{4}\). Plug in the values to get, \(sin(\frac{\pi}{6} + \frac{\pi}{4}) = sin(\frac{\pi}{6})cos(\frac{\pi}{4}) + cos(\frac{\pi}{6})sin(\frac{\pi}{4}) = \frac{1}{2} * \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} * \frac{\sqrt{2}}{2} = \frac{\sqrt{2} + \sqrt{6}}{4}\)

Step 3: Finding the Cosine Value

Use the cosine formula for the sum of two angles, i.e., \(cos(a+b) = cos(a)cos(b) - sin(a)sin(b)\). Again, \(a=\frac{\pi}{6}\) and \(b=\frac{\pi}{4}\). Plug in the values to get, \(cos(\frac{\pi}{6} + \frac{\pi}{4}) = cos(\frac{\pi}{6})cos(\frac{\pi}{4}) - sin(\frac{\pi}{6})sin(\frac{\pi}{4}) = \frac{\sqrt{3}}{2} * \frac{\sqrt{2}}{2} - \frac{1}{2} * \frac{\sqrt{2}}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\)

Step 4: Finding the Tangent Value

The tangent of an angle is the ratio of the sine to the cosine. So, \(tan(\frac{5 \pi}{12}) = \frac{sin(\frac{5 \pi}{12})}{cos(\frac{5 \pi}{12})} = \frac{\frac{\sqrt{2} + \sqrt{6}}{4}}{\frac{\sqrt{6} - \sqrt{2}}{4}} = - \sqrt{2} + \sqrt{6}\)

Key Concepts

Sine Sum Formula

Understanding the sine sum formula is crucial for solving trigonometry problems involving the sum of angles. The sine sum formula is given by
\(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
This equation allows us to find the sine of a sum of two angles by knowing the sines and cosines of these individual angles. For instance, to calculate the sine value of \(\frac{5 \pi}{12}\), we can express it as the sum of \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\), whose sine and cosine values are exact and well-known. By applying the sine sum formula, we quickly turn a complex problem into a simpler one, leveraging our knowledge of exact trigonometric values to find the result.

Cosine Sum Formula

Similarly, the cosine sum formula is an essential tool for finding the cosine of an angle formed by the sum of two other angles. The cosine sum formula is
\(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\).
Just as with the sine sum formula, this relation gives us a powerful technique to handle angles where the cosine value isn't directly known. By splitting the original angle into a sum of two angles with known cosine and sine values, we use the cosine sum formula to compute the cosine of the combined angle accurately. It aids in breaking down more complex problems into manageable calculations.

Tangent Ratio

The tangent of an angle is a fundamental ratio in trigonometry, defined as the ratio of the sine of the angle to its cosine
\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
Understanding this ratio is key to solving problems that require finding the tangent of an angle. When the sine and cosine values of an angle are known, their ratio gives its tangent. This concept is used to determine the tangent of the angle \(\frac{5 \pi}{12}\) in our problem, where we find the sine and cosine first, then their ratio to derive the tangent value. The tangent ratio is particularly useful because it simplifies the representation of an angle's slope in various applications.

Exact Trigonometric Values

Exact trigonometric values refer to the sine, cosine, and tangent values of specific angles that can be expressed as precise, rational numbers or simple radicals. Knowing these values is indispensable as they provide the building blocks for solving trigonometry problems. In the case of the angle \(\frac{5 \pi}{12}\), we use the known exact trigonometric values for \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\) to compute the required sine, cosine, and tangent values. Familiarity with these exact values is a significant advantage when dealing with trigonometric expressions and equations.