Problem 55 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$(\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta$$

Step-by-Step Solution

Answer

The equation is not an identity; it does not hold for all \(\theta\).

Step 1: Understand the Exercise

The problem asks us to determine whether the equation \((\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta\) is an identity. An identity must hold true for all values of \(\theta\). We will check this by expanding the left-hand side and comparing it with the right-hand side.

Step 2: Expand the Left-Hand Side

We start by expanding the left-hand side \((\sin \theta + \cos \theta)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\): \[(\sin \theta + \cos \theta)^2 = \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta\]

Step 3: Simplify Using an Identity

We know from the Pythagorean identity that \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, substituting back:\[\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = 1 + 2 \sin \theta \cos \theta\]

Step 4: Compare Both Sides

The right-hand side of the given equation is \(\sin^2 \theta + \cos^2 \theta = 1\). Comparing this with the expanded left-hand side, we have:\[1 + 2 \sin \theta \cos \theta eq 1\] for all non-zero values of \(\sin \theta \cos \theta\). This shows that the equation does not hold for all values of \(\theta\).

Step 5: Conclusion

Since the expanded left side does not equal the right side except when \(2 \sin \theta \cos \theta = 0\), the equation is not an identity. For example, at \(\theta = \frac{\pi}{4}\), the equation does not hold: \[(\sin(\frac{\pi}{4}) + \cos(\frac{\pi}{4}))^2 = 2 eq 1 = \sin^2(\frac{\pi}{4}) + \cos^2(\frac{\pi}{4})\]

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental concepts in trigonometry. It expresses a core relationship between the sine and cosine of an angle. Specifically, for any angle \(\theta\), the identity is represented as:\[\sin^2 \theta + \cos^2 \theta = 1\]This equation holds true for all values of \(\theta\). It is called an identity because it is universally valid, regardless of the specific angle used. The Pythagorean identity is derived from the Pythagorean theorem applied to a right triangle, where the hypotenuse equals 1 (the radius of the unit circle).Here’s why this identity is important:

• It provides a way to relate sine and cosine values to each other.
• It’s a useful tool for simplifying trigonometric expressions.
• It’s fundamental for proving other trigonometric identities.

Understanding this identity is crucial when solving or proving equations involving trigonometric functions. In our exercise, we used the Pythagorean identity to simplify and compare the expanded equations.

expanding binomials

Expanding binomials involves transforming an expression like \( (a + b)^2 \) into a more spread out expression. The formula to expand a binomial of the form \((a + b)^2\) is:\[(a + b)^2 = a^2 + 2ab + b^2\]This formula makes use of both addition and multiplication to break down the expression. Recall some key parts of expanding binomials:

• Square the first term: \(a^2\).
• Multiply the two terms together and double it: \(2ab\).
• Square the last term: \(b^2\).

In our given exercise, we started with the expression \((\sin\theta + \cos\theta)^2\). Applying the formula for binomial expansion, we converted it into:\[\sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta\]By expanding the binomial, we laid the groundwork for comparing the equations and determining that the original statement is not an identity.

trigonometric functions

Trigonometric functions describe the relationships between the angles and sides of triangles. The primary functions are sine, cosine, and tangent. These functions are essential in various areas of mathematics and practical applications, including physics, engineering, and more.For any angle \(\theta\):

• \(\sin \theta\) represents the ratio of the opposite side to the hypotenuse.
• \(\cos \theta\) represents the ratio of the adjacent side to the hypotenuse.
• \(\tan \theta\) is given by \(\frac{\sin \theta}{\cos \theta}\).

These functions are periodic and oscillate between -1 and 1. They are key to understanding concepts related to waves, cycles, and more.In our exercise, the focus was on \(\sin \theta\) and \(\cos \theta\). Their squared terms are part of the Pythagorean identity, and understanding their roles helped us in expanding the binomial and ultimately proving that the initial equation was not an identity.