Problem 63 - Algebra and Trigonometry

Question

Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) $$ \tan \alpha=\frac{3}{4}, \pi<\alpha<\frac{3 \pi}{2}, \text { and } \cos \beta=\frac{1}{4}, \frac{3 \pi}{2}<\beta<2 \pi $$

Step-by-Step Solution

Answer

The exact values are \(\cos (\alpha + \beta) = -\frac{3}{5}\), \(\sin (\alpha + \beta) = -\frac{\sqrt{15}}{5}\), and \(\tan (\alpha + \beta) = \sqrt{15}\).

Step 1: Find the sine and cosine values for alpha

The given value for tan alpha is \(\frac{3}{4}\). We can use the pythagorean identity to find sin alpha and cos alpha. We find the missing side of the right triangle to be \(\sqrt{1+ (\frac{3}{4})^2} = \frac{5}{4}\). So, \(\cos\alpha = \frac{4}{5}\) and \(\sin\alpha = -\frac{3}{5}\). The negative result comes from the quadrant of alpha.

Step 2: Find the sine and cosine values for beta

The given value for cos beta is \(\frac{1}{4}\). By finding the missing side of the right triangle, we get \(\sqrt{1- (\frac{1}{4})^2} = \frac{\sqrt{15}}{4}\). So, \(\cos\beta = \frac{1}{4}\) and \(\sin\beta = -\frac{\sqrt{15}}{4}\). The negative result comes from the quadrant of beta.

Step 3: Compute for cos(α + β)

By substituting the values of cos alpha, sin alpha, cos beta and sin beta into the identity \(\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta\), we get \(\cos(\alpha + \beta) = \frac{4}{5}*\frac{1}{4} - (-\frac{3}{5})*(-\frac{\sqrt{15}}{4}) = -\frac{3}{5}\).

Step 4: Compute for sin(α + β)

Using the values of cos alpha, sin alpha, cos beta and sin beta and the identity for sin(alpha + beta) which is \(\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta\), we get \(\sin(\alpha + \beta) = -\frac{3}{5}*\frac{1}{4} + \frac{4}{5}*(-\frac{\sqrt{15}}{4}) = -\frac{\sqrt{15}}{5}\).

Step 5: Compute for tan(α + β)

Using the definition of tan which is \(\tan x = \frac{\sin x}{\cos x}\), we get \(\tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} = \frac{-\sqrt{15}/5}{-3/5}= \sqrt{15}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that connects the sine and cosine of an angle. This identity is given by the equation:

• \( ext{sin}^2 \theta + ext{cos}^2 \theta = 1\)

This identity is derived from the Pythagorean Theorem applied to the unit circle, where the radius is 1. Using this identity can help us find one trigonometric function if we know the other. For example, if we know \( ext{tan} \alpha = \frac{3}{4}\), we can find \( ext{sin} \alpha\) and \( ext{cos} \alpha\). This involves recognizing that tan is the ratio of sine to cosine, \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\). We use the identity to establish the relationship:

• \(\sin^2 \alpha = 1 - \cos^2 \alpha\)

By setting up a right triangle with sides corresponding to the sine and cosine of \(\alpha\), we calculated \(\cos \alpha = \frac{4}{5}\) and \(\sin \alpha = -\frac{3}{5}\). This is crucial in solving for other identities like angle sum as we use these sine and cosine values.

Angle Sum Identities

Angle sum identities are essential trigonometric formulas for determining the sine, cosine, and tangent of the sum of two angles. They allow us to calculate these functions without directly measuring or knowing the angle. Here are the relevant angle sum identities:

• Cosine of angle sum: \[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]
• Sine of angle sum: \[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]

Using these formulas, we computed \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \) by substituting the known values of sine and cosine for both angles \(\alpha\) and \(\beta\). These identities simplify complex trigonometric calculations by breaking them down into fundamental operations involving known values.

Sine and Cosine Functions

Sine and cosine are the foundational trigonometric functions. They describe the relationship between the angles and lengths of a right triangle or the coordinates on the unit circle. These functions are periodic and repeat their values in a regular pattern with a period of \(2\pi\).

• Sine function (\(\sin\)): It gives the y-coordinate of a point on the unit circle and is defined as the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine function (\(\cos\)): It provides the x-coordinate on the unit circle and is the ratio of the adjacent side to the hypotenuse.

In the given problem, \(\cos \alpha\) and \(\cos \beta\) were provided, and \(\sin \alpha\) and \(\sin \beta\) were derived using these definitions alongside the Pythagorean identity. Understanding these functions and their relationships is crucial to applying them effectively in different trigonometric formulas.

Tangent Function

The tangent function is another key trigonometric ratio that can be expressed in terms of sine and cosine. It is defined as the ratio of the sine to the cosine of an angle:

• \[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

It represents the slope of the tangent line to the circle at a given point. In this exercise, the initial value \(\tan \alpha = \frac{3}{4}\) informed the determination of \(\sin \alpha\) and \(\cos \alpha\). Further in, when determining \(\tan(\alpha + \beta)\), we utilized the sine and cosine of the sum of angles to establish:

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

This substitution led to finding the exact tangent value for the angle sum, demonstrating how tangent is effectively a function of these two fundamental trigonometric components.