Problem 40 - Algebra and Trigonometry

Question

Suppose that \(\sin \alpha=\frac{3}{5}\) and \(\cos \beta=-\frac{12}{13}\) for quadrant II angles \(\alpha\) and \(\beta .\) Find the exact value of each of the following: a. \(\cos \alpha\) b. \(\sin \beta\) c. \(\cos (\alpha+\beta)\) d. \(\sin (\alpha+\beta)\) (Section \(6.2, \text { Example } 5)\)

Step-by-Step Solution

Answer

The answers are: a) \(\cos \alpha = -\frac{4}{5}\), b) \(\sin \beta = \frac{5}{13}\), c) \(\cos (\alpha+\beta)=\frac{16}{65}\), and d) \(\sin (\alpha+\beta)=-\frac{56}{65}\)

Step 1: Derive \(\cos \alpha\)

Since \(\alpha\) is a quadrant II angle and \(\sin \alpha = \frac{3}{5}\), the cosine of \(\alpha\) can be found by using the Pythagorean identity: \(\cos^{2}\alpha = 1 - \sin^{2}\alpha\). Taking the root of both sides (and remembering that cosine is negative in the second quadrant): \(\cos \alpha = - \sqrt{1 - \sin^{2}\alpha} = -\sqrt{1 - (\frac{3}{5})^{2}} = -\frac{4}{5}\)

Step 2: Evaluate \(\sin \beta\)

Similarly, since \(\beta\) is a quadrant II angle and we know that \(\cos \beta = -\frac{12}{13}\), we can use the Pythagorean identity to find \(\sin \beta\): \(\sin^{2}\beta = 1 - \cos^{2}\beta\). Computing for the square root of both sides produces: \(\sin \beta = \sqrt{1 - \cos^{2}\beta} = \sqrt{1 - (-\frac{12}{13})^{2}} = \frac{5}{13}\)

Step 3: Find \(\cos (\alpha+\beta)\)

Now, we use the formula \(\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\) to evaluate the cosine of the sum of the angles. Substituting the known values gives: \(\cos (\alpha + \beta) = -\frac{4}{5}(-\frac{12}{13}) -\frac{3}{5}\frac{5}{13} = \frac{16}{65} \)

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

Finally, using the formula \(\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\), the sine of the sum of the angles can be determined. Substitution yields: \(\sin (\alpha + \beta) = \frac{3}{5}(-\frac{12}{13}) -\frac{4}{5}\frac{5}{13} = -\frac{56}{65}\)

Key Concepts

Pythagorean identity

The Pythagorean identity is a cornerstone of trigonometry, deeply rooted in the Pythagorean theorem. It connects the sine and cosine of an angle in a trigonometric function. The identity is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \).

This formula is especially useful because it allows us to find one of these trigonometric functions when the other is known.

For example, if you know \( \sin \alpha \), you can find \( \cos \alpha \) even if the angle \( \alpha \) is not explicitly given. Just rearrange the identity like this: \( \cos^2 \alpha = 1 - \sin^2 \alpha \).

Take the square root (keeping in mind the angle's quadrant properties), and you have \( \cos \alpha = - \sqrt{1 - \sin^2 \alpha} \), since in the second quadrant cosine is negative.

cosine of sum of angles

The formula for the cosine of the sum of angles is pivotal in trigonometry. It allows the calculation of the cosine of two angles added together. The formula is: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).

Understanding how each component interacts provides deeper insight.

• When calculating \( \cos(\alpha + \beta) \), notice that you multiply the cosine of each angle together and subtract the product of their sines.
• This is not a simple addition or subtraction but a specific rule that combines these functions in a meaningful way.

Always substitute your calculated values of sine and cosine accurately. This keeps your calculations correct and aligns with the trigonometric identity properties.

sine of sum of angles

The sine of the sum of angles formula is equally essential as its cosine counterpart. This formula helps find the sine of a combined angle from two separate angles. It is expressed as \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \).

Here's a closer look at how it works:

• Both sine and cosine elements are present in the equation.
• The formula consists of addition rather than subtraction.
• Each term is the product of opposites: sine of one angle with cosine of the other.

One must insert the known values of \( \sin \alpha \), \( \cos \beta \), \( \cos \alpha \), and \( \sin \beta \) precisely to maintain alignment with trigonometrical identities.

second quadrant angles

Angles in the second quadrant, between \(90^\circ\) and \(180^\circ\), have distinctive properties in trigonometry.

Here's what you need to remember:

• Sine is positive in this quadrant, hence \( \sin \alpha = \frac{3}{5} \) makes sense.
• Cosine, on the other hand, is negative, which is why the solution for \( \cos \alpha \) using the Pythagorean identity resulted in a negative value.
• This distinction stems from the way a unit circle operates, influencing the sign of trigonometric functions based on the angle location.

Keeping this in mind lets you predict the behavior of each trigonometric function within specific quadrants without recalculating each characteristic from scratch.