Problem 37 - Thomas Calculus

Question

What happens if you take \(B=A\) in the trigonometric identity \(\cos (A-B)=\cos A \cos B+\sin A \sin B ?\) Does the result agree with something you already know?

Step-by-Step Solution

Answer

The result simplifies to the Pythagorean identity \(\cos^2 A + \sin^2 A = 1\), agreeing with known results.

Step 1: Identifying Given Identity

The trigonometric identity given is \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Our task is to investigate what happens when \(B = A\).

Step 2: Substitute B with A

Substitute \(B\) with \(A\) in the identity: \(\cos(A - A) = \cos A \cos A + \sin A \sin A\).

Step 3: Simplify the Expression

The expression simplifies to \(\cos(0) = \cos^2 A + \sin^2 A\) because \(A - A = 0\).

Step 4: Evaluate \(\cos(0)\)

Since \(\cos(0) = 1\), the equation becomes \(1 = \cos^2 A + \sin^2 A\).

Step 5: Combine with Known Identity

Recall the Pythagorean identity \(\cos^2 A + \sin^2 A = 1\). This confirms that when \(B = A\), the original identity simplifies to the well-known identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It states that for any angle \(A\), the equation \(\cos^2 A + \sin^2 A = 1\) holds true. This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. In this context, the hypotenuse becomes 1, and the legs are represented by \(\cos A\) and \(\sin A\).Understanding the Pythagorean identity can be very helpful. It acts as a bridge to connect trigonometric functions with basic geometric principles. When you solve problems involving trigonometric functions, knowing this identity will allow you to simplify expressions. It ensures that no matter what the angle \(A\) is, the sum of the squares of the sine and cosine will always equal 1. This makes it a powerful tool for verifying and deriving other identities.By using this identity, you can transition between trigonometric functions more easily. For example, if you know \(\sin A\), you can quickly find \(\cos A\) by rearranging the identity to \(\cos^2 A = 1 - \sin^2 A\). This conversion provides a simple way to move between sine and cosine, making calculations more straightforward.

Simplification in Trigonometry

Simplification in trigonometry often involves using known identities and mathematical properties to make expressions more manageable. In the given exercise, replacing \(B = A\) in the identity \(\cos(A - B) = \cos A \cos B + \sin A \sin B\) helps simplify it to the widely recognized Pythagorean identity.Let's break it down:

• When \(B = A\), the expression becomes \(\cos(0) = \cos^2 A + \sin^2 A\).
• Know that \(\cos(0) = 1\).
• This gives us the classic \(1 = \cos^2 A + \sin^2 A\).

In simplification exercises, such as this, identifying symmetry or components that can cancel out or equate to recognizable patterns can significantly reduce complexity.By exploring transformations like substituting variables or using fundamental identities such as the Pythagorean identity, trigonometric problems become less daunting and more intuitive to handle.

Trigonometric Functions

Trigonometric functions are mathematical functions related to the angles of triangles. The primary trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), each of which relates the angles of a right triangle to its side lengths.

• \(\sin A\) represents the ratio of the opposite side to the hypotenuse.
• \(\cos A\) represents the ratio of the adjacent side to the hypotenuse.
• \(\tan A\) is the ratio of \(\sin A\) to \(\cos A\).

These functions are crucial in not only solving geometric problems but also in analyzing wave patterns, oscillations, and various periodic phenomena.They are also central to forming various trigonometric identities, such as the Pythagorean identity. By using these functions, you can build a deeper understanding of relationships within a triangle and extend it to many fields within mathematics and science. Understanding how these functions interact with each other, and within identities, like the one in the exercise, reveals the structured beauty of trigonometry.