Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

(a) \(\frac{3}{5}\), (b) \(\frac{4}{5}\), (c) \(\frac{4}{25}\), (d) \(-\frac{117}{125}\), (e) \(\frac{44}{125}\), (f) \(-\frac{42}{25}\).

Step 1: Understand the Problem

Given are \(\tan \alpha = -\frac{7}{24}\) and \(\cot \beta = \frac{3}{4}\) with angle \(\alpha\) in the second quadrant and angle \(\beta\) in the third quadrant. You need to find the sine, cosine, and tangent of \(\alpha + \beta\) and \(\alpha - \beta\).

Step 2: Finding \\sin \alpha and \\cos \alpha

In the second quadrant, the sine is positive and cosine is negative. Use the identity \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\):\[\sin^2 \alpha + \cos^2 \alpha = 1\]\[\tan \alpha = -\frac{7}{24} \Rightarrow \sin \alpha = \frac{7}{\sqrt{7^2 + 24^2}} = \frac{7}{25}, \quad \cos \alpha = -\frac{24}{25}\]

Step 3: Finding \\sin \beta and \\cos \beta

In the third quadrant, both sine and cosine are negative. Use the identity \(\cot \beta = \frac{\cos \beta}{\sin \beta}\):\[\sin^2 \beta + \cos^2 \beta = 1\]\[\cot \beta = \frac{3}{4} \Rightarrow \sin \beta = -\frac{4}{5}, \quad \cos \beta = -\frac{3}{5}\]

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

Use the sine addition formula:\[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]\[= \frac{7}{25} \cdot \left(-\frac{3}{5}\right) + \left(-\frac{24}{25}\right) \cdot \left(-\frac{4}{5}\right)\]\[= -\frac{21}{125} + \frac{96}{125} = \frac{75}{125} = \frac{3}{5}\]

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

Use the cosine addition formula:\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]\[= \left(-\frac{24}{25}\right) \cdot \left(-\frac{3}{5}\right) - \frac{7}{25} \cdot \left(-\frac{4}{5}\right)\]\[= \frac{72}{125} + \frac{28}{125} = \frac{100}{125} = \frac{4}{5}\]

Step 6: Calculate \\tan(\\alpha+\\beta)

Use the tangent addition formula:\[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\]\[\tan \beta = \frac{1}{\cot \beta} = \frac{4}{3}\]\[= \frac{-\frac{7}{24} + \frac{4}{3}}{1 - \left(-\frac{7}{24}\right)\left(\frac{4}{3}\right)} = \frac{\frac{72}{72} - \frac{56}{72}}{1 + \frac{28}{72}} = \frac{\frac{16}{72}}{\frac{100}{72}} = \frac{4}{25}\]

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

Use the sine subtraction formula:\[\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\]\[= \frac{7}{25} \cdot \left(-\frac{3}{5}\right) - \left(-\frac{24}{25}\right) \cdot \left(-\frac{4}{5}\right)\]\[= -\frac{21}{125} - \frac{96}{125} = -\frac{117}{125}\]

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

Use the cosine subtraction formula:\[\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\]\[= \left(-\frac{24}{25}\right) \cdot \left(-\frac{3}{5}\right) + \frac{7}{25} \cdot \left(-\frac{4}{5}\right)\]\[= \frac{72}{125} - \frac{28}{125} = \frac{44}{125}\]

Step 9: Calculate \\tan(\\alpha-\\beta)

Use the tangent subtraction formula:\[\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\]\[\tan \beta = \frac{1}{\cot \beta} = \frac{4}{3}\]\[= \frac{-\frac{7}{24} - \frac{4}{3}}{1 + \left(-\frac{7}{24}\right)\left(\frac{4}{3}\right)} = \frac{-\frac{72}{72} - \frac{96}{72}}{1 + \frac{28}{72}} = \frac{-\frac{168}{72}}{\frac{100}{72}} = -\frac{42}{25}\]

Key Concepts

Addition Formulas

An essential idea in trigonometry is the use of addition formulas. These help us find the trigonometric values of the sum of two angles. For sine and cosine, the formulas are:

• \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
• \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \)

These formulas combine each individual angle's sine and cosine components. Breaking it down: the sine addition formula requires that each angle be multiplied by the cosine of the other and then added. Cosine, on the other hand, needs to subtract the product of each angle's sine values from the cosine product. The tangent addition formula is a bit trickier and combines tangents using addition in the numerator and a difference in a special product in the denominator.
Let's reflect on why these are useful. Employing these can transform complex angle problems into simpler calculations by leveraging known quantities. So if you know the sine and cosine of the angles involved, you can quickly calculate their combined effect in the trigonometric function.

Subtraction Formulas

Much like addition formulas, subtraction formulas play a key role in trigonometry. These simplify finding functions of angle differences. The formulas are as follows:

• \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)
• \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \)
• \( \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta} \)

Notice the fine differences with their addition counterparts. For sine subtraction, you subtract the cosine-sine product, unlike in addition where you add it.
Cosine subtraction swaps a minus sign for a plus, integrating both angles' sine. And the tangent subtraction is straightforward too – subtract the angles’ tangent in the numerator. These changes flip operation signs to accommodate the difference in angles rather than their sum.
Understanding and applying subtraction formulas opens up more accurate approaches in resolving angle-related trigonometric problems. Handy in instances involving determining unseen variables, these simplify complex scenarios to a much more manageable level, provided you have the requisite sine and cosine values at hand.

Angles in Quadrants

Understanding the quadrants is crucial in solving trigonometric problems. In this topic, angles are placed within one of four quadrants on a Cartesian plane. Each quadrant affects the sign (positive or negative) of the sine, cosine, and tangent functions:

• First Quadrant: All values are positive.
• Second Quadrant: Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, while sine and tangent are negative.

Keys to solving problems like our example involve recognizing this sign role and determining the exact trigonometric function's value. For instance, given that \( \tan \alpha = -\frac{7}{24} \) in the second quadrant, this aligns with cosine being negative. Conversely, \( \cot \beta = \frac{3}{4} \) in the third quadrant indicates sine and cosine as negative, consistent with the expected tangent positivity.
Grasping these quadrant traits assists not only in choosing correct identities but also validates calculated results against expected sign outcomes. This ensures solutions hold true in broader trigonometric context while adhering to universal function properties anchored by their respective quadrants.