Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

True or false: If the equation is not always true, give a counterexample. (a) \(\sin (A-B)=\sin (A)-\sin (B)\) (b) \(\cos (A+B)=\cos A+\cos B\)

Step-by-Step Solution

Answer

Both equations (a) and (b) are false. Counterexamples are: for (a) \((A, B) = (\frac{\pi}{2}, 0)\), for (b) \((A, B) = (\frac{\pi}{4}, \frac{\pi}{4})\).

Step 1: Checking Trigonometric Identity (a)

Let's consider \(A = B = \frac{\pi}{4}\). With these values, the left-hand side becomes \(\sin(0)=0\) whereas the right-hand side becomes \( \sin(\frac{\pi}{4})-\sin(\frac{\pi}{4}) = 0\). However, if we take \(A = \frac{\pi}{2}\) and \(B = 0\) then the left-hand side becomes \(\sin(\frac{\pi}{2}) = 1\), but the right-hand side becomes \( \sin(\frac{\pi}{2})-\sin(0) = 1\). Hence, (a) is not always true.

Step 2: Counterexample for Identity (a)

The counterexample can be \(A=\frac{\pi}{2}\), \(B=0\).

Step 3: Checking Trigonometric Identity (b)

Let's consider \(A = B = \frac{\pi}{4}\). With these values, left-hand side becomes \( \cos(\frac{\pi}{2})=0\) whereas the right-hand side becomes \( \cos(\frac{\pi}{4})+\cos(\frac{\pi}{4}) > 0 \). Hence, the identity does not hold true in this case.

Step 4: Counterexample for Identity (b)

The counterexample can be \(A=B=\frac{\pi}{4}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are an integral part of mathematics. They describe the relationships between the angles and sides of triangles. The most common trigonometric functions are sine, cosine, and tangent. Each of these functions has a specific role. For example:

• Sine gives the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine gives the ratio of the adjacent side to the hypotenuse.
• Tangent provides the ratio of the opposite side to the adjacent side.

Understanding these basic functions is crucial. They are widely used in solving equations and in many real-world applications. Moreover, they help in analyzing oscillatory phenomena, like sound waves. Trigonometric identities, which are equations involving these functions, allow us to simplify and solve trigonometric expressions.

Sine and Cosine

Sine and cosine are among the most fundamental trigonometric functions. They are particularly important in the context of circles and periodic functions. The term 'sine' originates from the Latin word "sinus," meaning a curve. Here are some key properties:

• Sine function: Periodic, with a period of \(2\pi\). It varies from -1 to 1. The sine wave originates at the origin \((0,0)\).
• Cosine function: Like sine, it's also periodic with the same range. The cosine wave starts at its maximum value of 1 when the angle is 0.

Both functions are closely related. The cosine function is essentially the sine function shifted by \(\frac{\pi}{2}\) radians. For example, \[\cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right)\]These functions are commonly found in the equations of waves and harmonic motion. They are essential for understanding the behavior of vibrating strings or electromagnetic fields.

Trigonometric Equations

Trigonometric equations involve using trigonometric functions to find unknown angles or values. Such equations are pivotal in mathematical modeling and problem-solving. In many situations, you need to verify whether an equation holds for all values or just specific ones. Consider equations like:

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

Checking these requires substitution of values into the equations. As shown in the given original exercise, certain identities hold only for specific values, not universally. This is where counterexamples come in handy. They demonstrate the conditions under which a given identity fails to apply. Solving these equations often involves various strategies, such as using known identities or algebraic manipulation. It helps confirm the validity of the trigonometric relationships and ensures they are used accurately in applications, such as engineering or physics.