Problem 45 - Contemporary Precalculus

Question

If \(x\) is in the first and \(y\) is in the second quadrant, \(\sin x=24 / 25,\) and \(\sin y=4 / 5,\) find the exact value of \(\sin (x+y)\) and \(\tan (x+y)\) and the quadrant in which \(x+y\) lies.

Step-by-Step Solution

Answer

Question: Determine the exact values of sin(x+y) and tan(x+y), and identify the quadrant in which the angle x+y lies, given that x is in the first quadrant, y is in the second quadrant, sin(x) = 24/25, and sin(y) = 4/5. Answer: The exact values for the trigonometric functions are sin(x+y) = -44/125 and tan(x+y) = 44/117. The angle x+y lies in the second quadrant.

Step 1: Apply the Sine Sum Formula

To find the exact value of sin(x+y), we will use the sine sum formula: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\). We already have \(\sin x = \frac{24}{25}\) and \(\sin y = \frac{4}{5}\) but we need to find \(\cos x\) and \(\cos y\).

Step 2: Calculate \(\cos x\) and \(\cos y\)

Since x is in the first quadrant, \(\cos x\) will be positive. We can use the Pythagorean identity \(\:\sin^2 x + \cos^2 x = 1\) to calculate \(\cos x\): \(\cos^2 x = 1 - \sin^2 x\) \(\cos x = \sqrt{1 - \left(\frac{24}{25}\right)^2}\) \(\cos x = \frac{7}{25}\) Since y is in the second quadrant, \(\cos y\) will be negative. We use the Pythagorean identity \(\:\sin^2 y + \cos^2 y = 1\) to calculate \(\cos y\): \(\cos^2 y = 1 - \sin^2 y\) \(\cos y = -\sqrt{1 - \left(\frac{4}{5}\right)^2}\) \(\cos y = -\frac{3}{5}\)

Step 3: Find the value of sin(x+y)

Now, substitute \(\sin x\), \(\sin y\), \(\cos x\), and \(\cos y\) into the sine sum formula to find sin(x+y): \(\sin(x+y) = \sin x \cos y + \cos x \sin y\) \(\sin(x+y) = \frac{24}{25} \cdot -\frac{3}{5} + \frac{7}{25} \cdot \frac{4}{5}\) \(\sin(x+y) = -\frac{72}{125} + \frac{28}{125}\) \(\sin(x+y) = -\frac{44}{125}\)

Step 4: Calculate \(\tan(x+y)\) using \(sin(x+y)\) and the sine and cosine sum formulas.

Now, we will find tan(x+y) using the formula \(\tan(x+y) = \frac{\sin(x+y)}{\cos x \cos y - \sin x \sin y}\). We have calculated sin(x+y) and have the values of the sine and cosine functions for x and y: \(\tan(x+y) = \frac{-\frac{44}{125}}{\frac{7}{25} \cdot -\frac{3}{5} - \frac{24}{25} \cdot \frac{4}{5}}\) \(\tan(x+y) = \frac{-44}{125} \cdot \frac{1}{\left( -\frac{21}{125} - \frac{96}{125}\right)}\) \(\tan(x+y) = \frac{-44}{( -117)} = \frac{44}{117}\)

Step 5: Determine in which quadrant x+y lies

The sum of an angle in the first quadrant (x) and an angle in the second quadrant (y) will be in the second quadrant. This is because their sum will be greater than 90° but less than 180°. Therefore, the angle x+y lies in the second quadrant.

Step 6: Final Results

The exact values of the trigonometric functions for the sum of the angles are: \(\sin(x+y) = -\frac{44}{125}\) \(\tan(x+y) = \frac{44}{117}\), and the angle (x+y) lies in the second quadrant.

Key Concepts

Sine Sum Formula

The Sine Sum Formula is a crucial tool in trigonometry, especially when dealing with the addition of angles. This formula is expressed as \( \sin(x+y) = \sin x \cos y + \cos x \sin y \). It allows us to find the sine of the sum of two angles using the sines and cosines of the individual angles.
For the problem at hand, we need to compute \( \sin(x+y) \). Given \( \sin x = \frac{24}{25} \) and \( \sin y = \frac{4}{5} \), we must first determine \( \cos x \) and \( \cos y \). By substituting these values, we can effectively use the Sine Sum Formula. This technique is essential for resolving the challenge of finding the sine of angle sums.

Pythagorean Identity

The Pythagorean Identity is a fundamental principle in trigonometry. It states that \( \sin^2 \theta + \cos^2 \theta = 1 \). With this identity, we can find one trigonometric function as long as we know the other.
For instance, when we have \( \sin x = \frac{24}{25} \), the identity helps draft \( \cos x \) by restructuring to \[ \cos^2 x = 1 - \sin^2 x \]. Simplifying further, \( \cos x = \frac{7}{25} \) in the first quadrant, where cosine is positive.
Similarly, knowing \( \sin y = \frac{4}{5} \) allows us to calculate \( \cos y \). In the second quadrant, where cosine is negative, we find \( \cos y = -\frac{3}{5} \). This identity is key in connecting sine and cosine through squared relationships.

Angle Sum Formula

The Angle Sum Formula relates to the addition of angles, useful for calculating values like \( \sin(x+y) \) and \( \tan(x+y) \). The cosine sum formula, also part of this category, is \( \cos(x+y) = \cos x \cos y - \sin x \sin y \). Even though it's not directly used in the calculation of \( \tan(x+y) \), it's implicitly involved.
To find \( \tan(x+y) \), we use \( \tan(x+y) = \frac{\sin(x+y)}{\cos x \cos y - \sin x \sin y} \). This formula neatly ties together our previous work with \( \sin(x+y) \) and uses both sine and cosine values obtained. These sum formulas are indispensable when working with angle addition in trigonometry.

Quadrant Determination

Determining the quadrant of an angle like \( x+y \) involves understanding the sum of angles. Since \( x \) is in the first quadrant (where both sine and cosine are positive) and \( y \) is in the second quadrant (where sine is positive and cosine is negative), we expect \( x+y \) to land in a different quadrant.
The angle sum \( x+y \) is evaluated by considering their range:

• An angle from the first quadrant is between 0° to 90°.
• An angle from the second is between 90° to 180°.

Therefore, adding these two angles results in an angle between 90° to 180°, placing \( x+y \) in the second quadrant. Understanding how quadrants work is key in predicting the behavior of angle sums in trigonometry.