Problem 15 - Precalculus with Limits

Question

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{17\pi}{12} = \dfrac{9\pi}{4} - \dfrac{5\pi}{6} \)

Step-by-Step Solution

Answer

The exact values of the sine, cosine, and tangent of the angle \( \dfrac{17\pi}{12} \) are \( \sin(\dfrac{17\pi}{12}) = -\dfrac{\sqrt{2} - \sqrt{6}}{4} \), \( \cos(\dfrac{17\pi}{12}) = -\dfrac{\sqrt{2} + \sqrt{6}}{4} \), and \( \tan(\dfrac{17\pi}{12}) = \dfrac{\sqrt{2} - \sqrt{6}}{\sqrt{2} + \sqrt{6}} \) respectively.

Step 1: Breakdown the Angle

The angle \( \dfrac{17\pi}{12} \) can be broken down into the sum of two angles whose sine and cosine we can easily determine, such as \( \pi/4 \) and \( \pi/6 \). Thus, \( \dfrac{17\pi}{12} = \dfrac{3\pi}{4} + \dfrac{\pi}{3} \).

Step 2: Find the Sine, Cosine, and Tangent

Use the sum of angles formulas for cosine and sine to find the exact values for the sine, cosine, and tangent of \( \dfrac{17\pi}{12} \). These are given by: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \), \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \) and \( \tan(a + b) = \dfrac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \). Use these formulas with \( a = \dfrac{3\pi}{4} \) and \( b = \dfrac{\pi}{3} \).

Step 3: Compute Exact Values

After substitution and simplification, you should get \( \sin(\dfrac{17\pi}{12}) = -\dfrac{\sqrt{2} - \sqrt{6}}{4} \), \( \cos(\dfrac{17\pi}{12}) = -\dfrac{\sqrt{2} + \sqrt{6}}{4} \), and \( \tan(\dfrac{17\pi}{12}) = \dfrac{\sqrt{2} - \sqrt{6}}{\sqrt{2} + \sqrt{6}} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that make it possible to simplify many complex expressions. These identities relate the angles and sides of triangles to trigonometric functions such as sine, cosine, and tangent. They remain true for all values of the involved variables. In this exercise, trigonometric identities help us solve angles expressed in radian form, like \( \dfrac{17\pi}{12} \), by breaking them into simpler, manageable parts.

Common identities include the Pythagorean identities like:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

These identities are not just algebraic facts; instead, they reflect the deep relationships between the angles and the corresponding values of sine, cosine, and tangent.

There are numerous other identities, like reciprocal identities \( \sec(\theta) = \frac{1}{\cos(\theta)} \), that help to find exact values of trigonometric functions, which is crucial for solving problems like calculating the exact trigonometric values of \( \dfrac{17\pi}{12} \).

Exact Trigonometric Values

Exact trigonometric values are the non-approximated values of trigonometric functions calculated for specific angles. Knowing these values by heart for common angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \) and so on is extremely helpful because they act as building blocks for finding exact trigonometric values of more complex angles.

For example:

• \( \sin(\frac{\pi}{6}) = \frac{1}{2} \)
• \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \tan(\frac{\pi}{3}) = \sqrt{3} \)

In the scenario described in the original question, the angle \( \dfrac{17\pi}{12} \) is decomposed to \( \dfrac{3\pi}{4} + \dfrac{\pi}{3} \) precisely because we can determine the sine and cosine for \( \frac{\pi}{4} \) and \( \frac{\pi}{3} \) using exact values, which simplifies the calculation. This method is often employed when the angle does not correspond to a specific memorized value, making it a strategic approach in trigonometry.

Angle Addition Formula

The angle addition formulas are essential tools in trigonometry used to find the sine, cosine, and tangent of sums or differences of angles. These formulas allow us to express trigonometric functions of composite angles in terms of the functions of simpler angles. They are particularly useful when the angle involved is not a standard angle.

Let's look at them step by step:

• The sine addition formula: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• The cosine addition formula: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• The tangent addition formula: \( \tan(a + b) = \dfrac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

In the exercise, by setting \( a = \dfrac{3\pi}{4} \) and \( b = \dfrac{\pi}{3} \), these formulas are used to precisely find the sine, cosine, and tangent values of \( \dfrac{17\pi}{12} \). This method is comprehensive because it uses known values to reach an exact answer, ensuring accuracy in calculations and fostering a deeper understanding of the trigonometric relationships.