Problem 58 - Precalculus Essentials

Question

Find the exact value of the following under the given conditions: (A) .\(\cos (\alpha+\beta)\) (B). \(\sin (\alpha+\beta)\) (C) \(\tan (\alpha+\beta)\) \(\sin \alpha=\frac{4}{5}, \alpha\) lies in quadrant \(I,\) and \(\sin \beta=\frac{7}{25}, \beta\) lies in quadrant II.

Step-by-Step Solution

Answer

(A) \(\cos (\alpha+\beta)\) = \(-\frac{472}{625}\), (B) \(\sin (\alpha+\beta)\) = \(-\frac{108}{625}\), (C) \(\tan (\alpha+\beta)\) = \(\frac{27}{118}\).

Step 1: Determine the Cosines

Since \(\sin \alpha=\frac{4}{5}\) and \(\alpha\) is in quadrant I (where cosine is positive), we have \(\cos \alpha = \sqrt{1-\sin^2 \alpha} = \frac{3}{5}\). Similarly, for \(\beta\), where \(\sin \beta=\frac{7}{25}\) and \(\beta\) is in quadrant II (where cosine is negative), we will have \(\cos \beta=-\sqrt{1-\sin^2\beta}=-\frac{24}{25}\).

Step 2: Calculate \(\cos(\alpha+\beta)\)

Now, we want to calculate \(\cos(\alpha+\beta)=\cos \alpha \cos \beta - \sin \alpha \sin \beta\). Therefore, plugging the values, we get \(\cos(\alpha+\beta) = \frac{3}{5} \times -\frac{24}{25} - \frac{4}{5}\times\frac{7}{25} = -\frac{72}{125}-\frac{112}{625} =-\frac{472}{625}\).

Step 3: Calculate \(\sin(\alpha+\beta)\)

Subsequently, use \(\sin(\alpha+\beta)=\sin \alpha \cos \beta + \cos \alpha \sin \beta\) to calculate the sine of \(\alpha + \beta\). Therefore, \(\sin(\alpha+\beta)=\frac{4}{5} \times -\frac{24}{25} + \frac{3}{5} \times \frac{7}{25} = -\frac{192}{625} + \frac{84}{625} = -\frac{108}{625}\).

Step 4: Calculate \(\tan(\alpha+\beta)\)

Finally, \(\tan(\alpha+\beta)=\frac{\sin(\alpha+\beta)}{\cos(\alpha+\beta)} = \frac{-\frac{108}{625}}{-\frac{472}{625}}=\frac{27}{118}\) which is the solution for \(\tan (\alpha + \beta)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding angles and their relationships in geometry. They include sine ( \( \sin \) ), cosine ( \( \cos \) ), and tangent ( \( \tan \) ). These functions relate the angles of a triangle to the lengths of its sides.

• Sine ( \( \sin \) ): This function gives the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine ( \( \cos \) ): It represents the ratio of the adjacent side to the hypotenuse.
• Tangent ( \( \tan \) ): This function expresses the ratio of the opposite side to the adjacent side.

In our exercise, \( \sin \alpha \) = \( \frac{4}{5} \) and \( \sin \beta = \frac{7}{25} \) . Using the Pythagorean identity, \( \cos^2 \alpha + \sin^2 \alpha = 1 \) , we find \( \cos \alpha \) = \( \frac{3}{5} \) for quadrant I and \( \cos \beta \) = -\( \frac{24}{25} \) for quadrant II.

Angle Addition Formulas

Angle addition formulas are essential tools in trigonometry. They allow us to calculate the sine, cosine, and tangent of the sum of two angles using their individual trigonometric values.

• Cosine Addition Formula: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• Sine Addition Formula: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Tangent Addition Formula: \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

In the exercise, the angles \( \alpha \) and \( \beta \) are given. Using the addition formulas, we've calculated \( \cos(\alpha + \beta) = -\frac{472}{625} \), \( \sin(\alpha + \beta) = -\frac{108}{625} \), and \( \tan(\alpha + \beta) = \frac{27}{118} \). These formulas reveal the combined effect of two angles on trigonometric functions.

Quadrant Analysis

Understanding quadrants is crucial in trigonometry for determining the signs of functions. The coordinate plane is divided into four quadrants, which influence the sign of trigonometric functions based on the angle's position.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

In this exercise, angle \( \alpha \) is in quadrant I, where both \( \sin \alpha \) and \( \cos \alpha \) are positive, hence \( \cos \alpha = \frac{3}{5} \). Angle \( \beta \) lies in quadrant II, which gives \( \cos \beta = -\frac{24}{25} \), as cosine is negative in this quadrant.
This analysis helps us predict and verify the signs of calculated trigonometric values.