Problem 52 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$\sqrt{\sin ^{2} t+\cos ^{2} t}=\sin t+\cos t$$

Step-by-Step Solution

Answer

The equation is not an identity, as it doesn't hold for all values of \(t\).

Step 1: Simplify the Left Side

First, simplify the left side of the equation. We have \(\sqrt{\sin^2 t + \cos^2 t}\). According to the Pythagorean identity, \(\sin^2 t + \cos^2 t = 1\). Therefore, \(\sqrt{\sin^2 t + \cos^2 t} = \sqrt{1} = 1\). So the left side simplifies to 1.

Step 2: Analyze the Right Side

Now, let's consider the right side of the equation \(\sin t + \cos t\). This expression does not simplify directly to 1 for all values of \(t\). The value of \(\sin t + \cos t\) can vary depending on \(t\).

Step 3: Compare Both Sides

Now we compare the simplified left side with the right side. After simplification, the left side of the equation becomes 1, while the right side remains \(\sin t + \cos t\). Since the right side can take on various values other than 1, the equation cannot hold for all values of \(t\).

Step 4: Validation with Specific Value

To validate, pick a specific example value for \(t\). For instance, if \(t = 0\), the left side is \(1\) and the right side is \(\sin 0 + \cos 0 = 0 + 1 = 1\), which seems true. However, pick another value like \(t = \frac{\pi}{4}\), the left side is still 1, but the right side is \(\sin \frac{\pi}{4} + \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}\). In this case, both sides are not equal.

Key Concepts

Trigonometric Functions

Trigonometric functions involve the study of angles and their relationships with side lengths in right-angled triangles. The main functions include sine, cosine, and tangent. Each of these functions takes an angle as input and outputs a ratio based on the geometry of a right triangle.
For instance, the sine function (\(\sin\theta\)) is calculated as the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function (\(\cos\theta\)) is the ratio of the adjacent side to the hypotenuse.
Let's delve deeper into sine and cosine, which are fundamental in expressing the Pythagorean identity. For any angle \(t\) in a right triangle:

• \(\sin^{2}t + \cos^{2}t = 1\)
• This identity holds true for any angle and is widely used in trigonometry to simplify expressions.
• In the equation from the exercise, a direct implication of this identity is \(\sqrt{\sin^2 t + \cos^2 t} = 1\), because the squared terms add to one.

Equation Validation

Equation validation is the process of confirming whether an equation holds true for all possible values of its variables. In this context, it involves verifying whether the equation \(\sqrt{\sin^{2}t+\cos^{2}t}=\sin t+\cos t\) is valid regardless of the value of \(t\).
The Pythagorean identity quickly simplifies the left side of this equation to 1. However, the right side, \(\sin t + \cos t\), does not have a static value and fluctuates based on the value of \(t\).
To check if the equation is universally true:

• Consider key values. For example, when \(t = 0\), the equation holds as both sides equal 1.
• However, for \(t = \frac{\pi}{4}\), the left remains 1, but the right becomes \(\sqrt{2}\), proving the inequality.
• Thus, the equation fails to be an identity because it does not maintain equality across the full range of \(t\).

Identity Proof

An identity in mathematics is an equation that is true for all possible values of its variables. Proving an identity involves demonstrating that both sides of an equation are equivalent under all conditions.
In the original exercise, the task was to determine if the equation \(\sqrt{\sin^{2}t+\cos^{2}t}=\sin t+\cos t\) serves as an identity. Start with basic simplifications to check both sides over the domain of \(t\).

• The left side simplifies to 1 using the Pythagorean identity: \(\sin^{2}t+\cos^{2}t=1\).
• The right side depends on the value of \(t\) and does not universally equal 1.
• Through testing different values, inconsistencies across both sides of the equation prove that it cannot be an identity.

Thus, rather than proving equality, such exercises often teach the limits of mathematical assumptions, guiding us to recognize when identities do or do not exist. This knowledge empowers deeper comprehension of the nature of mathematical equations.