Problem 58 - Algebra and Trigonometry

Question

Find the exact value of the following under the given conditions: a. \(\cos (\alpha+\beta)\) b. \(\sin (\alpha+\beta)\) c. \(\tan (\alpha+\beta)\) \(\sin \alpha=\frac{4}{5}, \alpha\) lies in quadrant I, and \(\sin \beta=\frac{7}{25}, \beta\) lies in quadrant II.

Step-by-Step Solution

Answer

a. \(\cos (\alpha+\beta) = -\frac{33}{125}\) b. \(\sin (\alpha+\beta) = -\frac{63}{125}\) c. \(\tan (\alpha+\beta) = \frac{63}{33} or \frac{21}{11}\)

Step 1: Find the values of Cosine

First, identify the values of \(\cos \alpha\) and \(\cos \beta\). Use the Pythagorean identity \(\cos^2(x) = 1 - \sin^2(x)\). Since \(\sin \alpha = \frac{4}{5}\) and \(\alpha\) is in the first quadrant where cosine is positive, \(\cos \alpha = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \frac{3}{5}\). Similarly for \(\beta\), as \(\sin \beta = \frac{7}{25}\) and \(\beta\) is in the second quadrant where cosine is negative, \(\cos \beta = -\sqrt{1 - \left(\frac{7}{25}\right)^2} = -\frac{24}{25}\).

Step 2: Calculate \(\cos (\alpha+\beta)\)

Using the trigonometric identity \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\), calculate \(\cos(\alpha + \beta)\). Insert the found values: \(\cos(\alpha + \beta) = \frac{3}{5} \cdot -\frac{24}{25} - \frac{4}{5} \cdot \frac{7}{25} = -\frac{33}{125}\).

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

Using the trigonometric identity \(\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\), calculate \(\sin(\alpha + \beta)\). Insert the found values: \(\sin(\alpha + \beta) = \frac{4}{5} \cdot -\frac{24}{25} + \frac{3}{5} \cdot \frac{7}{25} = -\frac{63}{125}\).

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

Lastly, use the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) to calculate \(\tan(\alpha + \beta)\). Insert the found values: \(\tan(\alpha + \beta) = \frac{-\frac{63}{125}}{-\frac{33}{125}} = \frac{63}{33}\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a cornerstone in the wonderful world of trigonometry. It states that for any angle, the square of the sine plus the square of the cosine equals one: \[ \sin^2(x) + \cos^2(x) = 1.\] This elegant formula works for all angles, demonstrating the inherent relationship between the sine and cosine functions. Just like in our exercise, we used this identity to find the cosine values, knowing the sine values of the angles. It's helpful to remember that in a right-angled triangle, the Pythagorean Theorem relates the lengths of the sides, and in a similar way, the Pythagorean identity relates the squares of the trigonometric functions of an angle.

Sine and Cosine

Sine and cosine are the bread and butter of trigonometric functions. They are essential for analyzing right-angled triangles and circular motion. The sine of an angle, typically represented as \(\sin(x)\), is the ratio of the opposite side to the hypotenuse in a right-angled triangle. On the other hand, the cosine of an angle, designated as \(\cos(x)\), is the ratio of the adjacent side to the hypotenuse. The values of these functions are key to solving trigonometric problems and can be found using the unit circle or trigonometric tables for exact angles. As in the exercise, when we know the sine value and the quadrant in which an angle resides, we can determine the cosine value using the Pythagorean identity.

Angle Addition Formulas

Angle addition formulas offer a way to construct the sine, cosine, or tangent of a sum or difference of two angles from the sines and cosines of those angles individually. The formulas look like this:\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta,\] and\[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.\] These are like recipes that allow us to mix together the trigonometric flavors of different angles, similar to how we combined the sine and cosine of \(\alpha\) and \(\beta\) in the provided exercise to find the cosine and sine of their sum. Understanding and applying these formulas are crucial for solving trigonometric equations and for complex wave interference problems in physics.

Exact Trigonometric Values

Exact trigonometric values refer to the precise sine, cosine, and tangent values for certain commonly used angles like 0°, 30°, 45°, 60°, and 90°, amongst others. These values are often memorized or quickly accessible through a unit circle or special triangles, such as the 45-45-90 and 30-60-90 triangles. When solving trigonometric equations, like the one in our exercise, finding the exact values instead of decimal approximations keeps the solution clean and accurate. Mastery of these exact values is beneficial not only for solving homework problems but also for standardized tests where calculators may not be permitted, and understanding how they are derived can deepen one's comprehension of trigonometry as a whole.