Problem 57 - Precalculus Essentials

Question

Find the exact value of the following under the given conditions: (A) .\(\cos (\alpha+\beta)\) (B). \(\sin (\alpha+\beta)\) (C) \(\tan (\alpha+\beta)\) \(\sin \alpha=\frac{3}{5}, \alpha\) lies in quadrant \(1,\) and \(\sin \beta=\frac{3}{13}, \beta\) lies in quadrant II.

Step-by-Step Solution

Answer

The exact value of \(\cos(\alpha+\beta)\) is -\frac{61}{65}, of \(\sin(\alpha+\beta)\) is -\frac{21}{65}, and of \(\tan(\alpha+\beta)\) is -\frac{21}{61}

Step 1: Find Cosine Value

Since \(\alpha\) is in the first quadrant, the cosine of \(\alpha\) is positive and can be calculated using the Pythagorean theorem, \(\cos \alpha= \sqrt{1-\sin^{2} \alpha}=\sqrt{1-(\frac{3}{5})^2}=\frac{4}{5}\). Similarly, since \(\beta\) lies in the second quadrant, \(\cos \beta\) is negative and its value is \(\cos \beta= -\sqrt{1-\sin^{2} \beta}=-\sqrt{1-(\frac{3}{13})^2}=-\frac{12}{13}\).

Step 2: Apply Cosine Sum Formula

The sum formula for cosine is \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\). By substituting the calculated values and given values, we get \(\cos (\alpha+\beta) = (\frac{4}{5}*(-\frac{12}{13})) - (\frac{3}{5}*\frac{3}{13}) = -\frac{61}{65}\).

Step 3: Apply Sine Sum Formula

The sum formula for sine is \(\sin(\alpha + \beta) =\sin \alpha \cos \beta +\cos \alpha \sin \beta\). Substituting the calculated and given values, we get \(\sin (\alpha+\beta) = (\frac{3}{5}*(-\frac{12}{13})) + (\frac{4}{5}*\frac{3}{13}) = -\frac{21}{65}\).

Step 4: Calculate the Tangent

The tangent function is the ratio of the sine to the cosine: \(\tan (\alpha+\beta) = \frac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)}\). Substituting the calculated values, we get \(\tan (\alpha+\beta) = -\frac{21}{61}\).

Key Concepts

Cosine Sum Formula

The cosine sum formula is an essential trigonometric identity used to find the cosine of the sum of two angles. It is particularly useful in situations where you know the sine and cosine values of the individual angles but need to determine the cosine of their sum.
For any two angles \( \alpha \) and \( \beta \), the cosine sum formula is given by:

• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

To apply this formula, it is important to know the individual cosine and sine values of each angle. In the problem, these values were calculated as follows:

• \( \cos \alpha = \frac{4}{5} \), because \( \alpha \) lies in the first quadrant where all trigonometric functions are positive.
• \( \cos \beta = -\frac{12}{13} \), because \( \beta \) is in the second quadrant where cosine is negative.
• \( \sin \alpha = \frac{3}{5} \) and \( \sin \beta = \frac{3}{13} \) are provided.

By substituting these values into the cosine sum formula, you can find \( \cos(\alpha + \beta) \). Remember to multiply the associated sine and cosine values together carefully, being mindful of the signs, as these impact the final result. For the exercise, the computation gives us:

• \( \cos(\alpha + \beta) = (-\frac{61}{65}) \)

Sine Sum Formula

The sine sum formula is another key trigonometric identity that helps find the sine of the sum of two angles. This formula is particularly handy when you have the sine and cosine values of each angle and need to calculate the sine of their sum.
For any two angles \( \alpha \) and \( \beta \), the formula is:

• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)

To use the formula, we need the cosine and sine values of \( \alpha \) and \( \beta \). We already found these in the previous section:

• \( \cos \alpha = \frac{4}{5} \), \( \sin \alpha = \frac{3}{5} \)
• \( \cos \beta = -\frac{12}{13} \), \( \sin \beta = \frac{3}{13} \)

By inserting these values into the sine sum formula, we can determine \( \sin(\alpha + \beta) \). Care must be taken to maintain the sign of the terms involved, particularly since \( \cos \beta \) is negative for this exercise. Calculating carefully results in:

• \( \sin(\alpha + \beta) = -\frac{21}{65} \)

Tangent Sum Formula

The tangent sum formula provides a way to calculate the tangent of the sum of two angles. This formula can be incredibly useful when working with trigonometric exercises and identities. It is expressed as the ratio of the sine and cosine of the sum of the angles.
The formula for the tangent of the sum of two angles \( \alpha \) and \( \beta \) is:

• \( \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \)

This formula requires the computed sine and cosine values from the sum formulas described earlier. Using the previously calculated values:

• \( \sin(\alpha + \beta) = -\frac{21}{65} \)
• \( \cos(\alpha + \beta) = -\frac{61}{65} \)

You can determine \( \tan(\alpha + \beta) \) by directly dividing these expressions, always being mindful of the algebraic signs to avoid errors. Therefore, in this scenario, we have:

• \( \tan(\alpha + \beta) = -\frac{21}{61} \)