Problem 64 - Precalculus Essentials

Question

Find the exact value of the following under the given conditions: (A) .\(\cos (\alpha+\beta)\) (B). \(\sin (\alpha+\beta)\) (C) \(\tan (\alpha+\beta)\). 64\. \(\sin \alpha=\frac{5}{6}, \frac{\pi}{2}<\alpha<\pi,\) and \(\tan \beta=\frac{3}{7}, \pi<\beta<\frac{3 \pi}{2}\)

Step-by-Step Solution

Answer

\(\cos (\alpha + \beta) = \frac{1}{3}\), \(\sin (\alpha + \beta) = -\frac{4}{3}\) and \(\tan (\alpha + \beta) = -4\).

Step 1 - Find the Cosine and Sine values for α

We are given that \(\sin \alpha=\frac{5}{6}\) and that \(\frac \pi 2 < \alpha < \pi\), which means α is in the second quadrant where sine is positive and cosine is negative. We can find the cosine using the Pythagorean identity: \(\cos^2 \alpha = 1 - \sin^2 \alpha\). Substituting the given value, we get \(\cos^2 \alpha = 1 - (\frac{5}{6})^2 = \frac{1}{36}\). Because cosine is negative in the second quadrant, we have \(\cos \alpha= -\sqrt{\frac{1}{36}} = -\frac{1}{6}\).

Step 2 - Find the Cosine and Sine values for β

We are given that \(\tan \beta=\frac{3}{7}\) and that \(\pi < \beta < \frac {3\pi}{2}\), which means β is in the third quadrant where both cosine and sine are negative. Using the Pythagorean identity in the form \(\tan^2 \beta = \sin^2 \beta / \cos^2 \beta\), you can solve for \(\cos^2 \beta = \sin^2 \beta / \tan^2 \beta\). You can find \(\sin \beta\) by applying the formula: \(\sin \beta = - \cos \beta \cdot \tan \beta\), due to both sine and cosine being negative in the third quadrant, the negative signs will cancel, and we have \(\sin \beta= \frac{3}{\sqrt{58}}, \cos \beta= -\frac{7}{\sqrt{58}}\).

Step 3 - Apply the Sum Formulas

Now we can easily find the trigonometric functions of α + β by substituting the derived sine and cosine values for α and β to the sum formulas: \(\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\), \(\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\) and \(\tan (\alpha + \beta) = \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)}\). Plugging in the values gives \(\cos (\alpha + \beta)= \frac{1}{3}, \sin (\alpha + \beta) = -\frac{4}{3}\), and \(\tan (\alpha + \beta) = -4\).

Key Concepts

Sum and difference formulas

To solve problems involving the sum of two angles, it's crucial to understand the sum formulas:

• The cosine sum formula: \[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]
• The sine sum formula: \[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]
• The tangent sum formula: \[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\]

These formulas help find the exact values of trigonometric functions for specific conditions, like in the exercise where you find \(\cos(\alpha+\beta)\), \(\sin(\alpha+\beta)\), and \(\tan(\alpha+\beta)\).Remember to calculate each angle's trigonometric parts first, as shown in the step-by-step process. Substitute these parts into the respective formulas to evaluate the results step by step.

Exact values in trigonometry

Exact values are fundamental in trigonometry as they allow us to express functions without decimal approximation. In the exercise, you calculated exact values by using known trigonometric identities. For example, given \( \sin \alpha = \frac{5}{6}\), utilize the Pythagorean identity:\[\cos^2 \alpha = 1 - \sin^2 \alpha\]This determines that \(\cos \alpha = -\frac{1}{6}\) due to the quadrant.Exact values let us work precisely with equations and avoid errors related to rounding. When functions are expressed in radicals or fractions, like \(\tan \beta = \frac{3}{7}\), the reasoning follows similarly. Calculate other needed functions systematically to maintain precision. These values will directly fit into the sum formulas, ensuring coherent final results without approximation errors.

Trigonometric functions in quadrants

Recognizing where angles lie on the unit circle is vital for evaluating trigonometric functions correctly. This exercise involved angles \(\alpha\) in the second quadrant \((\frac{\pi}{2} < \alpha < \pi)\) and \(\beta\) in the third quadrant \((\pi < \beta < \frac{3\pi}{2})\).
Each quadrant affects:

• The signs of trigonometric functions:
  • In the second quadrant, sine is positive, while cosine and tangent are negative.
  • In the third quadrant, both sine and cosine are negative, making tangent positive.
• How identities are applied, e.g., using negative or positive square roots.

Understanding quadrants ensures accurate function evaluations and prevents sign errors, which are common mistakes in such problems.