Problem 66 - Precalculus with Limits

Question

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{7\pi}{12} \)

Step-by-Step Solution

Answer

The exact values for the sine, cosine, and tangent of \( \frac{7\pi}{12} \) are \( \sin(\frac{7\pi}{12}) = \sqrt{2 - \sqrt{3}}/2 \), \( \cos(\frac{7\pi}{12}) = \sqrt{2 + \sqrt{3}}/2 \) and \( \tan(\frac{7\pi}{12}) = 2 - \sqrt{3} \) respectively.

Step 1: Express the Angle as Half of a Recognizable Angle

Express the angle \( \frac{7\pi}{12} \) as half of a recognizable angle. This can be done by expressing \( \frac{7\pi}{12} \) as \( \frac{1}{2} \times \frac{7\pi}{6} \). The trigonometric values of \( \frac{7\pi}{6} \) are known.

Step 2: Apply the Half-Angle Formulas

The half-angle formulas are given by: \n\n1. \( \sin(\frac{\theta}{2}) = \pm \sqrt {\frac {1 - \cos \theta}{2}} \)\n2. \( \cos(\frac{\theta}{2}) = \pm \sqrt {\frac {1 + \cos \theta}{2}} \)\n3. \( \tan(\frac{\theta}{2}) = \pm \sqrt {\frac {1 - \cos\theta}{1 + \cos\theta}} \) or \( \frac {1 - \cos\theta}{ \sin\theta}\)\n\nWe know that for \( \frac{7\pi}{6} \):\n\n1. \( \sin(\frac{7\pi}{6}) = - \frac {1}{2} \)\n2. \( \cos(\frac{7 \pi}{6}) = - \sqrt{3}/2\)\n\nSubstitute these into the half angle formulas to get the sine, cosine and tangent of \( \frac{7\pi}{12} \). The choice of positive or negative square root in the half-angle formula depends on the quadrant of the original angle \( \frac{7\pi}{6} \).

Step 3: Simplification

Simplify the expressions to get the final exact values of the sine, cosine, and tangent of \( \frac{7\pi}{12} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that allow us to find the values of trigonometric functions like sine, cosine, and tangent, using known angles. They are essential for solving trigonometry problems, especially when dealing with non-standard angles. The half-angle formulas are a crucial type of identity that help us express the sine, cosine, or tangent of half an angle in terms of the trigonometric functions of the original angle. These formulas can be particularly useful for angles that are not found on the common unit circle positions.

• The half-angle formulas for sine, cosine, and tangent are:
  • \[ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}} \]
  • \[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}} \]
  • \[ \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} \quad \text{or} \quad \frac{1 - \cos\theta}{\sin\theta} \]
• Choice of sign (positive or negative) is based on the quadrant in which the halved angle is located.

Understanding these identities helps you apply them appropriately and simplify complex trigonometric expressions.

Angle Conversion

Converting angles from one form to another is an essential skill in trigonometry. Angles can be given in various units or expressed in different ways, which might necessitate conversion for calculations. In the exercise, we have the angle \( \frac{7\pi}{12} \), which at first may not seem very familiar.To simplify working with this angle, we express it as half of a well-known angle. Our exercise converts \( \frac{7\pi}{12} \) into \( \frac{1}{2} \times \frac{7\pi}{6} \). In this way, recognizing that \( \frac{7\pi}{6} \) is equivalent to \( 210^\circ \) and part of known trigonometric values, makes our calculations more straightforward. Understanding how to break down angles using multiplication or division helps leverage more familiar angles and allows for the correct application of trigonometric identities. Practicing these conversions can greatly enhance problem-solving efficiency in trigonometry.

Exact Values in Trigonometry

Exact values in trigonometry refer to the numerical values that precisely represent trigonometric functions for specific angles. Rather than approximate decimal values, we use expressions involving integers, fractions, and square roots that hold true within a perfect mathematical context.Using identities like the half-angle formulas, we can determine these exact values for angles not typically listed in tables or unit circles. In our exercise, we are tasked with finding the exact values of \( \sin\), \( \cos\), and \( \tan\) for the angle \( \frac{7\pi}{12} \) using the known exact values of \( \frac{7\pi}{6} \).For \( \frac{7\pi}{6} \):

• \( \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} \)
• \( \cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2} \)

Substituting into the half-angle formulas and selecting the appropriate sign based on the quadrant provides the exact values for \( \frac{7\pi}{12} \). Recognizing patterns in exact values helps build intuition for solving diverse trigonometry exercises with ease.