Problem 37 - Thomas Calculus

Question

What happens if you take \(B=A\) in the 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

Taking \(B = A\) confirms the identity \(\cos^2 A + \sin^2 A = 1\).

Step 1: Substitute B with A

Replace all instances of \(B\) with \(A\) in the given trigonometric identity, \(\cos(A-B) = \cos A \cos B + \sin A \sin B\). This yields \(\cos(A-A) = \cos A \cos A + \sin A \sin A\).

Step 2: Simplify the Left Side

Calculate \(\cos(A-A)\). Since \(A-A = 0\), this simplifies to \(\cos(0)\). We know from trigonometric identities that \(\cos(0) = 1\). So, the left side equals 1.

Step 3: Simplify the Right Side

On the right side, \(\cos A \cos A = \cos^2 A\) and \(\sin A \sin A = \sin^2 A\). Combine these to form \(\cos^2 A + \sin^2 A\).

Step 4: Apply the Pythagorean Identity

Recognize that \(\cos^2 A + \sin^2 A = 1\), which is a well-known Pythagorean identity.

Step 5: Equate Both Sides

Since both sides equal 1, the equation holds true: \(1 = 1\).

Key Concepts

Cosine Function

In trigonometry, the cosine function is a fundamental concept often symbolized as \( \cos \). It is an essential building block in understanding wave patterns, rotations, and angles, particularly in unit circles. The cosine function reflects the x-coordinate of a point as it moves around a unit circle, created by rotating a ray around the origin. This characteristic provides it with a periodic nature, repeating every \(2\pi\) radians or 360 degrees.
To visualize, imagine a right-angled triangle. The cosine of an angle \(\theta\) in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. Mathematically, it is expressed as:

• \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

A few important properties of the cosine function include:

• **Periodicity:** It repeats every \(2\pi\) radians.
• **Symmetry:** It is an even function, meaning \( \cos(-x) = \cos(x) \).
• **Range:** The values of \( \cos(x) \) range from -1 to 1.

Understanding the cosine function is key to navigating through various trigonometric identities and calculations you'll encounter further in this topic.

Pythagorean Identity

The Pythagorean Identity is one of the cornerstones of trigonometry. It provides a critical relationship between the sine and cosine functions. This identity stems from the Pythagorean Theorem which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In trigonometric terms, it's expressed as:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

This identity confirms that for any angle \(\theta\), the sum of the squares of the sine and cosine will always be 1. This is particularly powerful because it allows for simplifications in trigonometric equations.In practice:

• When you need to find \(\cos \theta\) or \(\sin \theta\), knowing one allows you to find the other using this identity.
• It is widely used to verify other trigonometric identities and simplify expressions as seen in the original exercise solution.

Using the Pythagorean Identity efficiently helps in transforming and solving complex trigonometric problems with ease.

Angle Addition Formulas

The Angle Addition Formulas are critical tools in trigonometry that allow for the calculation of trigonometric functions at the sum of two angles. These formulas provide a method for breaking down complicated trigonometric angles into manageable calculations. For cosine, the formula is:

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

This formula helps in determining the cosine of an angle composed of the sum of two other angles. But it's not just limited to addition; subtraction is equally essential, which is demonstrated by:

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

These formulas are useful for:

• Calculating unknown angle values from known values.
• Simplifying complex expressions in both calculus and coordinate geometry.
• Proving other trigonometric identities.

In the exercise, substituting \(B\) with \(A\) in the subtraction formula led us directly into an application of the Pythagorean Identity, showcasing how these tools can overlap and interlink in solving trigonometric problems.