Problem 6 - Algebra and Trigonometry

Question

Use the formula for the cosine of the difference of two angles to solve Exercises \(1-12\) In Exercises \(5-8,\) each expression is the right side of the formula for \(\cos (\alpha-\beta)\) with particular values for \(\alpha\) and \(\beta\) a. Identify \(\alpha\) and \(\beta\) in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. $$ \cos 50^{\circ} \cos 5^{\circ}+\sin 50^{\circ} \sin 5^{\circ} $$

Step-by-Step Solution

Answer

The values are \(\alpha = 50^{\circ}\) and \(\beta = 5^{\circ}\). The expression can be written as \(\cos (50^{\circ}-5^{\circ})\) and its exact value is \(\frac{\sqrt{2}}{2}\).

Step 1: Identify Alpha and Beta

In the term \( \cos 50^{\circ} \cos 5^{\circ}+\sin 50^{\circ} \sin 5^{\circ} \), \(\alpha = 50^{\circ}\) and \(\beta = 5^{\circ}\) according to the formula for a cosine of difference of two angles.

Step 2: Write as Cosine of an Angle

Rewrite the term using the formula for cosine of the difference of two angles, i.e., \(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\) to get \(\cos (50^{\circ}-5^{\circ}) = \cos 50^{\circ} \cos 5^{\circ}+\sin 50^{\circ} \sin 5^{\circ}\)

Step 3: Find Exact Value

The exact value of the expression \(\cos (50^{\circ}-5^{\circ}) = \cos 45^{\circ}\), which is \(\frac{\sqrt{2}}{2}\).

Key Concepts

Trigonometric Identities

In trigonometry, identities are formulas that hold true for all possible values of the involved variables. One such important identity is the cosine of the difference of two angles, given by \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \). This identity is particularly useful as it allows us to transform expressions involving trigonometric functions of angles into simpler or alternative forms. For example, in the provided exercise, we successfully rewrote an expression, \( \cos 50^{\circ} \cos 5^{\circ} + \sin 50^{\circ} \sin 5^{\circ} \), using this identity. This equation became \( \cos(50^{\circ} - 5^{\circ}) \), streamlining our calculation process. Such transformations are valuable when finding precise trigonometric values or simplifying complex expressions.
Recognizing and applying these identities can greatly enhance problem-solving efficiency. The cosine of the difference identity, in particular, is a common tool for dealing with angle manipulations, which is why it's a staple in trigonometry classes.

Angle Values

Understanding angle values is crucial when working with trigonometric functions. Angles in trigonometry can be measured in degrees or radians, with 360 degrees equating to \(2\pi\) radians. The exercise discusses angles like \(50^{\circ}\) and \(5^{\circ}\), which are typical degree measurements used in trigonometric computations. When dealing with problems, it's essential to not only identify the angles involved but to understand how to manipulate them effectively.
In the highlighted problem, knowing the angle values facilitated the use of the cosine identity for the difference of two angles. By accurately identifying \( \alpha = 50^{\circ} \) and \( \beta = 5^{\circ} \), we could rewrite the expression into a more familiar form, \( \cos(45^{\circ}) \). This angle manipulation underscores the importance of carefully analyzing and handling angle values in trigonometry to achieve correct results.
Additionally, angles like \( \45^{\circ} \) often have widely recognized exact trigonometric values known by heart, which brings us to the next topic.

Exact Trigonometric Values

Exact trigonometric values refer to the precise values of trigonometric functions at particular angles. Trigonometry relies on these known values for efficient problem solving, as they eliminate the need for approximation. For example, at \(45^{\circ}\), we know that \( \cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \). This exactness is critical when solving problems, as illustrated in the exercise where we ended up with \( \cos(45^{\circ}) \).
Having these values memorized can significantly speed up calculations and improve accuracy. Other common angles with exact trigonometric values include \(0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), among others.
Recognizing and using these exact values simplifies the process of solving trigonometric identities and expressions. By reducing the reliance on calculators for approximate values, students can deepen their understanding of the relationships and patterns in trigonometric functions.