Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

If \(\alpha\) and \(\beta\) are acute angles such that \(\cos \alpha=\frac{4}{5}\) and \(\tan \beta=\frac{8}{15}\), find (a) \(\sin (\alpha+\beta)\) (b) \(\cos (\alpha+\beta)\) (c) the quadrant containing \(\alpha+\beta\)

Step-by-Step Solution

Answer

(a) \(\frac{77}{85}\), (b) \(\frac{36}{85}\), (c) first quadrant.

Step 1: Finding \( \sin \alpha \) using Pythagorean Identity

Since \( \alpha \) is an acute angle and \( \cos \alpha = \frac{4}{5} \), we use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \). Therefore, \( \sin^2 \alpha = 1 - \left( \frac{4}{5} \right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \). Hence, \( \sin \alpha = \frac{3}{5} \) since \( \alpha \) is acute and \( \sin \alpha \) is positive.

Step 2: Finding \( \cos \beta \) and \( \sin \beta \) using \( \tan \beta \)

Given \( \tan \beta = \frac{8}{15} \), if we let the opposite side be 8 and the adjacent side be 15, the hypotenuse can be found using the Pythagorean theorem: \( \text{hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = 17 \). Thus, \( \cos \beta = \frac{15}{17} \) and \( \sin \beta = \frac{8}{17} \) for the acute angle \( \beta \).

Step 3: Calculate \( \sin(\alpha + \beta) \) using Angle Addition Formula

The formula for \( \sin(\alpha + \beta) \) is \( \sin \alpha \cos \beta + \cos \alpha \sin \beta \). Plugging in, we get \( \sin \alpha \cos \beta = \frac{3}{5} \times \frac{15}{17} = \frac{9}{17} \) and \( \cos \alpha \sin \beta = \frac{4}{5} \times \frac{8}{17} = \frac{32}{85} \). Adding these gives \( \sin(\alpha + \beta) = \frac{9}{17} + \frac{32}{85} = \frac{77}{85} \).

Step 4: Calculate \( \cos(\alpha + \beta) \) using Angle Addition Formula

The formula for \( \cos(\alpha + \beta) \) is \( \cos \alpha \cos \beta - \sin \alpha \sin \beta \). This becomes \( \cos \alpha \cos \beta = \frac{4}{5} \times \frac{15}{17} = \frac{12}{17} \) and \( \sin \alpha \sin \beta = \frac{3}{5} \times \frac{8}{17} = \frac{24}{85} \). Therefore, \( \cos(\alpha + \beta) = \frac{12}{17} - \frac{24}{85} = \frac{36}{85} \).

Step 5: Determine the quadrant of \( \alpha + \beta \)

Since both \( \alpha \) and \( \beta \) are acute, their sum \( \alpha + \beta \) will be less than 180 degrees (\( \pi \) radians), meaning it can lie in either the first quadrant (where both sine and cosine are positive) or second quadrant (sine is positive, cosine is negative). Given \( \sin(\alpha + \beta) = \frac{77}{85} \) and \( \cos(\alpha + \beta) = \frac{36}{85} \), both are positive, indicating \( \alpha + \beta \) is in the first quadrant.

Key Concepts

Angle Addition Formulas

The angle addition formulas are pivotal in trigonometry for calculating the trigonometric functions of the sum or difference of two angles. For any angles \( \alpha \) and \( \beta \), the sine and cosine of their sum can be found using:

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

These formulas are incredibly useful when \( \sin \), \( \cos \), or \( \tan \) of \( \alpha \) and \( \beta \) are known. They allow us to derive the trigonometric values of complex angle expressions without directly measuring them. In this particular example, these formulas helped us find the values of \( \sin(\alpha + \beta) \) and \( \cos(\alpha + \beta) \) easily by substituting known values obtained from the Pythagorean theorem and tangent's definition.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that for any angle \( \theta \), the square of its sine plus the square of its cosine equals one:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is derived from the Pythagorean theorem applied to the unit circle, where the hypotenuse is 1. In this exercise, we used the Pythagorean identity to find \( \sin \alpha \) from \( \cos \alpha \), knowing \( \alpha \) is an acute angle. Since \( \alpha \) is acute, \( \sin \alpha \) must be positive, which simplifies our calculation to select the positive root.

Acute Angles

An acute angle is any angle that is greater than 0 degrees but less than 90 degrees. In trigonometry, this is important because it dictates the sign and the range of the trigonometric functions:

• In acute angles, the sine, cosine, and tangent functions are all positive.

This property is why we were able to confidently assume \( \sin \alpha \) was positive without further ado. Knowing \( \alpha \) and \( \beta \) are acute also informed our calculation that their sum, \( \alpha + \beta \), could potentially be another acute angle, thus remaining positive for sine and cosine values.

Quadrants

The coordinate plane can be divided into four quadrants, which help determine the sign of the trigonometric functions based on the angle’s position. Here's a quick breakdown of the quadrants:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, and cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, and cosine is positive.

In this problem, finding that both \( \sin(\alpha + \beta) \) and \( \cos(\alpha + \beta) \) are positive allowed us to conclude that \( \alpha + \beta \) is in the first quadrant. This systematic approach using quadrant analysis is essential for verifying the angle’s legitimacy when applying trigonometric identities.