Problem 37 - Algebra 2

Question

Which One Does Belong? Identify the equation that does not belong with the other three. Explain your reasoning. $$ \begin{array}{ll}{\sin ^{2} \theta+\cos ^{2} \theta=1} & {1+\cot ^{2} \theta=\csc ^{2} \theta} \\ {\sin ^{2} \theta-\cos ^{2} \theta=2 \sin ^{2} \theta} & {\tan ^{2} \theta+1=\sec ^{2} \theta}\end{array} $$

Step-by-Step Solution

Answer

\( \sin^2 \theta - \cos^2 \theta = 2 \sin^2 \theta \) does not belong because it's not a known trigonometric identity.

Step 1: Identify Common Trigonometric Identities

Review the given equations to identify which ones are standard trigonometric identities. We have the following identities:- \( \sin^2 \theta + \cos^2 \theta = 1 \) is a Pythagorean identity.- \( 1 + \cot^2 \theta = \csc^2 \theta \) is another Pythagorean identity.- \( \tan^2 \theta + 1 = \sec^2 \theta \) is also a Pythagorean identity.

Step 2: Analyze the Remaining Equation

Examine the equation \( \sin^2 \theta - \cos^2 \theta = 2 \sin^2 \theta \). This equation does not simplify to any known trigonometric identity and appears to be different from the others.

Step 3: Compare the Equations

Compare all the equations: the first, second, and fourth equations are known Pythagorean trigonometric identities, while the third equation does not fit this pattern and cannot be rewritten into any standard form like the others.

Step 4: Determine the Equation That Doesn't Belong

Conclude that the equation \( \sin^2 \theta - \cos^2 \theta = 2 \sin^2 \theta \) does not belong with the others because it does not conform to a standard trigonometric identity and lacks the structural relation seen in Pythagorean identities.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. This identity is based on the Pythagorean theorem, which relates the sides of a right-angled triangle to its hypotenuse. In the context of trigonometric functions, the most well-known Pythagorean identity is:

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

This identity is akin to saying that the square of the sine of an angle plus the square of the cosine of that angle always equals one. It holds true for any angle \( \theta \).
Another form of Pythagorean identity includes:

• \( 1 + \cot^2 \theta = \csc^2 \theta \)
• \( \tan^2 \theta + 1 = \sec^2 \theta \)

These identities are extensions of the basic sine and cosine identity and can be derived by dividing the original identity by \( \cos^2 \theta \) or \( \sin^2 \theta \). They provide a foundational tool for manipulating and solving various trigonometric equations.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, or tangent. Solving these equations often requires applying various algebraic techniques and trigonometric identities like the Pythagorean identities.
In the exercise at hand, we are tasked to identify an equation that does not belong with others in terms of being a recognizable trigonometric identity. Recognizable trigonometric identities follow fixed patterns and relationships between trigonometric functions.
The equations given were:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)
• \( \tan^2 \theta + 1 = \sec^2 \theta \)
• \( \sin^2 \theta - \cos^2 \theta = 2 \sin^2 \theta \)

The first, second and fourth equations are recognized as Pythagorean identities. The third equation, however, does not fit known trigonometric equations because it cannot be rewritten as a standard identity. Hence, understanding and identifying these equations is crucial in providing the correct classification.

Sine and Cosine Functions

The sine and cosine functions are among the most fundamental trigonometric functions. They are periodic functions with a period of \( 2\pi \), and they describe the relationship between the angles and lengths in a right-angled triangle. Their values oscillate between -1 and 1.
The sine function, \( \sin \theta \), gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. The cosine function, \( \cos \theta \), provides the ratio of the adjacent side to the hypotenuse.
These functions are related through the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity expresses a fundamental relationship found in all right-angled triangles, where the sum of the squares of the two sides is equal to the square of the hypotenuse when the sides are normalized (each divided by the hypotenuse).
Understanding sine and cosine, and their derived identities, is essential for solving many trigonometric equations and problems across mathematics and physics.