Problem 5 - Precalculus Essentials

Question

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. $$\cos 50^{\circ} \cos 20^{\circ}+\sin 50^{\circ} \sin 20^{\circ}$$

Step-by-Step Solution

Answer

The angles are \(\alpha = 50^{\circ}\) and \(\beta = 20^{\circ}\). The expression can be given as \(\cos (50^{\circ} - 20^{\circ}) = \cos 30^{\circ}\). The exact value of the expression is \(\sqrt{3}/2\).

Step 1: Identification of angles

The given expression is \(\cos 50^{\circ} \cos 20^{\circ}+\sin 50^{\circ} \sin 20^{\circ}\). Here, the angle \(\alpha\) is 50 degrees and \(\beta\) is 20 degrees.

Step 2: Write the expression as cosine of an angle

Using the identity for cosine of difference of two angles (\(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\)), and now replacing \(\alpha\) with 50 degrees and \(\beta\) with 20 degrees, the expression \(\cos 50^{\circ} \cos 20^{\circ}+\sin 50^{\circ} \sin 20^{\circ}\) can be written as \(\cos (50^{\circ} - 20^{\circ}) = \cos 30^{\circ}\).

Step 3: Find the exact value

Now find the exact value of \(\cos 30^{\circ}\) using the standard values of trigonometric functions. The standard value of cosine of 30 degrees is \(\sqrt{3}/2\).

Key Concepts

Cosine of Difference

The cosine of difference formula is a fundamental trigonometric identity. It's expressed as: \[ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \] This identity helps us to rewrite expressions into more manageable forms. Whenever you see a combination of cosine and sine in a specific way, they're often hinting at this identity.

• If you have an expression such as \( \cos(50^\circ) \cos(20^\circ) + \sin(50^\circ) \sin(20^\circ) \), you should recognize its similarity to the cosine of difference formula right away.
• This tells you that this expression equals \( \cos(50^\circ - 20^\circ) \), which is simply \( \cos(30^\circ) \).

Recognizing cosine of difference expressions is crucial in simplifying trigonometric problems, making it easier to solve them. Once you know the identity, you can quickly turn multiple terms into an angle cosine.

Exact Trigonometric Values

Understanding exact trigonometric values simplifies solving many math problems. These values are derived from key angles in the unit circle like \(30^\circ, 45^\circ,\) and \(60^\circ\).

• For example, \( \cos(30^\circ) \) equals \( \sqrt{3}/2 \) and this is a well-known result that frequently appears in exercises.
• Other standard exact values you might need are \( \sin(30^\circ) = 1/2 \), \( \cos(45^\circ) = \sqrt{2}/2 \), and \( \sin(60^\circ) = \sqrt{3}/2 \).

Always memorize these core values because they come in handy for calculations in exams and real-world applications. Having these numbers on hand allows you to confidently tackle expressions, computations, or simplifications without hesitation.

Angle Identification

Identifying angles correctly is crucial in using trigonometric identities. In problems like our example, angle recognition helps convert the expression into simpler forms using identities.

• First, look at the given expression \( \cos 50^\circ \cos 20^\circ + \sin 50^\circ \sin 20^\circ \). Notice that the angles, 50 degrees and 20 degrees, match up with those found in the cosine of difference formula.
• Here, we recognize \( \alpha = 50^\circ \) and \( \beta = 20^\circ \). Once identified, these angles can be used within the identity, turning the expression into \( \cos(50^\circ - 20^\circ) \).

Appropriately identifying and substituting angles are your first steps before applying trigonometric identities. Doing so ensures accuracy and makes complex expressions manageable.