Problem 20 - Algebra 2

Question

Mental Math Find the value of each trigonometric expression. $$ \cos 183^{\circ} \cos 93^{\circ}+\sin 183^{\circ} \sin 93^{\circ} $$

Step-by-Step Solution

Answer

The value of the given trigonometric expression is 0.

Step 1: Recognition of Angle Addition Identity

We notice that the given expression matches the form of cosine of the sum of two angles (A and B) which is \(\cos(A + B) = \cos A \cos B - \sin A \sin B\). But in our case, there is a '+' instead of a '-'. Hence, we can modify our expression to match this identity.

Step 2: Converting the Expression

We can write our expression in terms of cosine of difference of two angles because \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Now, this matches with the given expression, so we can represent \(\cos 183^{\circ} \cos 93^{\circ} + \sin 183^{\circ} \sin 93^{\circ}\) as \(\cos(183^{\circ} - 93^{\circ})\)

Step 3: Simplify and Solve

The expression simplifies to \(\cos 90^{\circ}\), and the value of this expression is 0 as cos90° equals 0

Key Concepts

Cosine Difference Identity

Trigonometric identities are useful tools for simplifying complex expressions, and the cosine difference identity is one of these vital mathematical instruments. It allows us to rewrite an expression involving angles in such a way that it becomes easier to solve. The cosine difference identity is represented as:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

This identity shows us how to express the cosine of the difference between two angles in terms of their individual trigonometric functions. In this exercise, recognizing and applying this identity was crucial in simplifying the original trigonometric expression.
The expression \( \cos 183^{\circ}\cos 93^{\circ} + \sin 183^{\circ} \sin 93^{\circ} \) matches the cosine difference identity, allowing us to represent it as \( \cos(183^{\circ} - 93^{\circ}) \). Understanding this identity is essential for both simplifying expressions and solving trigonometric problems efficiently.

Mental Math

Mental math involves performing calculations in your head without the use of written computations or calculators. This skill is particularly helpful when dealing with trigonometric identities, as it allows for quick simplifications and solutions.
In our exercise, mental math can be leveraged if you are familiar with the critical angles and their cosine and sine values. For example, knowing that the cosine of \(90^{\circ}\) is 0 can instantly solve the final expression without any deep calculations.

• Familiarize yourself with trigonometric values of standard angles like \(30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}\), etc.
• Practice identifying patterns and identities like the cosine difference to see solutions quickly.

Over time, practicing these skills will improve your ability to perform such calculations mentally, saving time and effort during exams and practical applications.

Angle Addition Identity

Another key trigonometric identity related to our original exercise is the angle addition identity. Although not directly used in the current example, understanding this identity can aid in larger frameworks of trigonometric problem-solving. The cosine of an angle addition is given by:

• \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Recognizing the structure of trigonometric expressions as either sum or difference enables simplification. While our exercise involved the cosine of a difference, the additory angle identity is similarly structured and often paired with the cosine difference identity.
These identities provide mathematical elegance and simplicity, enabling us to break down and grasp more complex trigonometric problems. A sound understanding of both the cosine difference and addition identities is invaluable for tackling a wide range of questions in trigonometry, algebra, and calculus.