Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Match with each of the expressions on the right with the equivalent expression on the left. (a) \(\cos ^{2} x\) (i) \(1-\cos ^{2} x\) (b) \(\sin ^{2} x\) (ii) \(\quad 1-\sin ^{2} x\) (c) \(\tan ^{2} x\) (iii) \(\frac{1}{2}-\frac{1}{2} \cos 2 x\) (d) \(\sin x\) (iv) \(-1+\sec ^{2} x\) (e) \(\cos x\) (v) \(\quad 1-\sin ^{2}(x+\pi / 2)\) (f) \(-\cos x\) (vi) \(\sin (x-\pi / 2)\) (vii) \(\frac{1}{2}+\frac{1}{2} \cos 2 x\) (viii) \(\cos (x-\pi / 2)\)

Step-by-Step Solution

Answer

The matched pairs are (a-ii), (b-i), (c-iv), (d-viii), (e-v), (f-vi).

Step 1: Analyze Left Side Expressions and their Identities

Apply the relevant identities on the left side: (a) \(\cos ^{2} x\), (b) \(\sin ^{2} x\), (c) \(\tan ^{2} x\), (d) \(\sin x\), (e) \(\cos x\), and (f) \(-\cos x\). Using the identities, we can conclude: (a) \(1-\sin ^{2} x\) (b) \(1-\cos ^{2} x\) (c) \(-1+\sec ^{2} x\) (d) \(\cos (x-\pi / 2)\) (e) \(\sin (x+\pi / 2)\) (f) \(-\sin (x+\pi / 2)\)

Step 2: Matching with Right Side Expressions

Let's now match our results from Step 1 with the right side expressions: (a) \(\cos ^{2} x\) -> (ii) \(1-\sin ^{2} x\), (b) \(\sin ^{2} x\) -> (i) \(1-\cos ^{2} x\), (c) \(\tan ^{2} x\) -> (iv) \(-1+\sec ^{2} x\), (d) \(\sin x\) -> (viii) \(\cos (x-\pi / 2)\), (e) \(\cos x\) -> (v) \(\sin (x+\pi / 2)\), (f) \(-\cos x\) -> (vi) \(-\sin (x+\pi / 2)\)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It relates the squares of the sine and cosine functions to the number one. The classic expression of the identity is: \[ \sin^2 x + \cos^2 x = 1 \] This could be re-arranged based on what's needed:

• \( \cos^2 x = 1 - \sin^2 x \)
• \( \sin^2 x = 1 - \cos^2 x \)

These re-arrangements show how we can express one function in terms of the other based on completeness. In the context of the original exercise, the expression \(1 - \sin^2 x\) is equivalent to \(\cos^2 x\). This is a direct application of the Pythagorean Identity, making it an essential tool for solving trigonometric expressions.
Using this identity helps simplify problems because it allows us to move between different trigonometric expressions without changing their value.

Angle Sum and Difference Identities

These identities are also critical in trigonometry. They allow us to find the sine, cosine, or tangent of an angle that is expressed as a sum or difference of two other angles. For example, the cosine of a difference is expressed as:\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] Similarly, the sine of a difference is:\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] These can be used to simplify and understand the given functions in the problem like \(\cos(x-\pi/2)\) and \(\sin(x-\pi/2)\). For instance:

• \(\cos(x-\pi/2) = \sin x\)
• \(\sin(x-\pi/2) = -\cos x\)

Using these identities, we can effectively transform trigonometric expressions to match equivalent forms. This plays a key role in matching expressions, as seen in the exercise where these identities make connections between forms obvious.

Trigonometric Transformation

Trigonometric transformations involve altering expressions to simplify equations or expressions. This could mean changing the position of angles or expressing functions in different forms using fundamental identities. For example, changing \(-\cos x\) to \(-\sin(x+\pi/2)\) uses transformations and the angle sum and difference identities.
Transformations often involve shifting angles by \(\frac{\pi}{2}\), \(\pi\), or other convenient values. This shift can lead to simple transformations such as:

• \(-\cos x\) can also be expressed as \(-\sin(x+\pi/2)\) through transformation.
• \(\sin(x+\pi/2) = \cos x\)

Mastery of these transformations is crucial for simplifying trigonometric expressions. They are also necessary for finding equivalents, as seen in the original exercise where expressions were transformed to find matching equivalents from the list.

Functions

Functions in trigonometry often refer to sine, cosine, tangent, and their reciprocal functions. Each has specific properties and transforms over different ranges:

• **Sine Function**: Periodic with a period of \(2\pi\), it starts at zero and oscillates between -1 and 1.
• **Cosine Function**: Shares the same period and range limitations as sine but is shifted by \(\frac{\pi}{2}\).
• **Tangent Function**: It has a period of \(\pi\) and ranges from negative to positive infinity, crossing zero at integer multiples of \(\pi\).

Understanding these functions is crucial as they form the basis of all transformations and identities. They can be manipulated and combined using identities to simplify and solve problems in trigonometry, as demonstrated in matching various expressions. Choosing the correct function or transformation is the key to unlocking the correct equivalent expression.