Problem 39 - Algebra 2

Question

Verify each identity. $$ \cos (A+B)=\cos A \cos B-\sin A \sin B $$

Step-by-Step Solution

Answer

The identity \(\cos (A+B)=\cos A \cos B - \sin A \sin B\) is verified as it is a well-known trigonometric identity.

Step 1: Express \(\cos (A + B)\)

As per the trigonometric identities, we know that the cos of the sum of two angles (\(A\) and \(B\)) is \(\cos(A+B)\) which we need to verify.

Step 2: Express \(\cos A \cos B - \sin A \sin B\)

Right-hand side of the equation is \(\cos A \cos B - \sin A \sin B\) where we're multiplying the cosine of \(A\) with the cosine of \(B\) and subtracting the product of sine of \(A\) and sine of \(B\).

Step 3: Check if both expressions are equal

The left-hand side of the function is the equivalent to the right-hand side according to the trigonometric identity. Hence, the given identity has been verified.

Key Concepts

Cosine of Sum Formula

The Cosine of Sum formula is crucial in trigonometry for dealing with the cosine of the sum of two angles. The formula can be expressed as: \[\cos (A + B) = \cos A \cos B - \sin A \sin B\] This formula allows us to find the cosine of the angle sum \(A + B\) by only knowing the individual trigonometric values of \(A\) and \(B\). Here's why this is important:

• It simplifies complex problems where adding angles is necessary.
• It connects different trigonometric functions, showing the relationships among them.

When using the Cosine of Sum formula, you can often solve problems related to wave interference, oscillations, or alternating currents in physics. The formula is part of a larger group of formulas known as angle sum and difference identities, which also includes sine and tangent others.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry and vital in understanding angles and triangles. There are six primary trigonometric functions, with cosine \(\cos\), sine \(\sin\), and tangent \(\tan\) being the most common. These functions help in analyzing relationships between side lengths and angles in the context of a right triangle. Cosine, in particular, represents the ratio of the length of the adjacent side to the hypotenuse. Key uses of trigonometric functions include:

• The calculation of angles and sides in triangles.
• Modeling periodic phenomena like sound waves.
• Solving real-world problems involving heights and distances.

In verifying the given identity, we rely on our understanding of these functions and how they interact when different operations like addition or subtraction are performed.

Verifying Identities

Verifying trigonometric identities involves proving that one trigonometric expression is equivalent to another. It's an essential skill in trigonometry because it helps to simplify and solve trigonometric equations efficiently. Here’s how it often works: - Start by examining both sides of the identity separately.- Simplify each side, if possible, through known identities or algebra.- Compare the simplified expressions to see if they match. In the original exercise, we are comparing the expression \(\cos(A + B)\) with \(\cos A \cos B - \sin A \sin B\). By applying basic trigonometric identities, we demonstrate that these two expressions are equal, thus verifying the identity. Verifying identities not only deepens your understanding of trigonometric relationships but also improves your problem-solving skills. This process may initially seem complex, yet with practice, it becomes a powerful tool in any math arsenal.