Problem 22 - Precalculus: Functions and Graphs

Question

If \(\tan \alpha=-\frac{7}{24}\) and \(\cot \beta=\frac{3}{4}\) for a second- quadrant angle \(\alpha\) and a third-quadrant angle \(\boldsymbol{\beta},\) find (a) \(\sin (\alpha+\beta)\) \(\langle b\rangle \cos (\alpha+\beta)\) \(\langle c) \tan (\alpha+\beta)\) (d) \(\sin (\alpha-\beta)\) (e) \(\cos (\alpha-\beta)\) (f) \(\tan (\alpha-\beta)\)

Step-by-Step Solution

Answer

Calculate trigonometric identities using given angles' characteristics.

Step 1: Determine sine and cosine for \(\alpha\)

Given that \(\tan \alpha = -\frac{7}{24}\) and \(\alpha\) is in the second quadrant, use the identity \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\). In the second quadrant, \(\sin \alpha > 0\) and \(\cos \alpha < 0\). Assume \(\cos \alpha = -24x\) and \(\sin \alpha = 7x\), resulting in \(\frac{7x}{-24x} = -\frac{7}{24}\). Hence, \(x^2 = \frac{1}{\underline{\phantom{xx}}} (24^2 + 7^2)\). Solve for \(x\) and determine \(\sin \alpha\) and \(\cos \alpha\).

Step 2: Determine sine and cosine for \(\beta\)

Given that \(\cot \beta = \frac{3}{4}\) and \(\beta\) is in the third quadrant, use the identity \(\cot \beta = \frac{\cos \beta}{\sin \beta}\). In the third quadrant, both \(\sin \beta\) and \(\cos \beta\) are negative. Assume \(\cos \beta = -3y\) and \(\sin \beta = -4y\), resulting in \(\frac{-3y}{-4y} = \frac{3}{4}\). The hypotenuse \(r = \sqrt{3^2 + 4^2} = 5\), so \(\cos \beta = -\frac{3}{5}\) and \(\sin \beta = -\frac{4}{5}\).

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

Use the identity \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\). Substitute the values: \(\sin \alpha = \frac{7}{25}\), \(\cos \alpha = -\frac{24}{25}\), \(\cos \beta = -\frac{3}{5}\), and \(\sin \beta = -\frac{4}{5}\). Calculate the result for \(\sin (\alpha + \beta)\).

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

Use the identity \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\). Substitute the known values and compute \(\cos (\alpha + \beta)\).

Step 5: Calculate \(\tan (\alpha + \beta)\)

Use the identity \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\). Identify \(\tan \beta = \frac{4}{3}\), and substitute into the expression. Simplify to find \(\tan (\alpha + \beta)\).

Step 6: Calculate \(\sin (\alpha - \beta)\)

Use the identity \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\). Substitute the known values and calculate \(\sin (\alpha - \beta)\).

Step 7: Calculate \(\cos (\alpha - \beta)\)

Use the identity \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\). Substitute the known values and compute \(\cos (\alpha - \beta)\).

Step 8: Calculate \(\tan (\alpha - \beta)\)

Use the identity \(\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\). Substitute \(\tan \alpha = -\frac{7}{24}\) and \(\tan \beta = \frac{4}{3}\) into the expression. Simplify to find \(\tan (\alpha - \beta)\).

Key Concepts

Angle Addition Identity

The Angle Addition Identity is a fundamental concept in trigonometry that helps us find the sine, cosine, and tangent of the sum of two angles. When dealing with trigonometric expressions like \(\sin(\alpha + \beta)\), \(\cos(\alpha + \beta)\), or \(\tan(\alpha + \beta)\), this identity comes into play. Here's what you need to remember:

• For sine: \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\).
• For cosine: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\).
• For tangent: \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\).

These formulas allow you to combine the angles and calculate their respective trigonometric values in one go.
A big hint here is to carefully identify the known values of \(\sin\), \(\cos\), and \(\tan\) for each angle before plugging them into these formulas.

Angle Subtraction Identity

The Angle Subtraction Identity is similar to the Angle Addition Identity but is used for finding trigonometric values when subtracting angles. It is just as useful and often required alongside its addition counterpart.

• For sine: \(\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\).
• For cosine: \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\).
• For tangent: \(\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\).

Using these identities involves recognizing when to use subtraction and understanding the signs will change for the sine and tangent identities. A well-calculated solution relies on proper substitution of calculated values such as \(\sin \alpha\) and \(\cos \beta\) which are specific to the given quadrants.

Trigonometric Functions

Trigonometric functions form the backbone of trigonometry and are essential in solving problems involving angles. The key functions include sine, cosine, and tangent, and each has a unique ratio derived from a right-angled triangle or on the unit circle. Here's a quick look at these functions:

• \(\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}\)
• \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
• \(\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{\sin \theta}{\cos \theta}\)

These functions are influenced by the angle's position in a specific quadrant, which affects their signs. Knowing these basic ratios helps in deducing identities for compound angles and their calculation.

Quadrants of the Unit Circle

The unit circle is a crucial tool in trigonometry that helps determine the trigonometric values of angles, particularly their sign. It is divided into four quadrants, each with distinct characteristics based on the signs of sine and cosine values of angles:

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

For calculations, knowing the quadrant helps ensure that the correct signs are applied to the trigonometric functions. For example, in the second quadrant, \(\tan \alpha\) would be negative due to cosine being negative while sine is positive, exactly as noted in the exercise. This concept is vital for correctly applying trigonometric identities, as the sign can directly impact calculated results in compounded angles.