Problem 1 - Algebra 2 and Trigonometry

Question

Is \(\sin \theta=\sqrt{1-\cos ^{2} \theta}\) an identity? Explain why or why not.

Step-by-Step Solution

Answer

No, it's not an identity because it ignores the possible negative value of \( \sin \theta \).

Step 1: Understand the Equation

The equation given is \( \sin \theta = \sqrt{1 - \cos^2 \theta} \). We want to determine whether this equation is an identity. An identity is an equation that is true for all values of the variable (in this case, \( \theta \)).

Step 2: Use Pythagorean Identity

There is a known trigonometric identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Rearrange this identity to express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \): \( \sin^2 \theta = 1 - \cos^2 \theta \).

Step 3: Relate to Given Equation

Notice that if \( \sin^2 \theta = 1 - \cos^2 \theta \), then \( \sin \theta \) should be ±\( \sqrt{1 - \cos^2 \theta} \). The expression \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) ignores the negative root possibility.

Step 4: Conclusion on the Identity

Since \( \sin \theta \) can be both positive and negative depending on the quadrant in which \( \theta \) is located, the equation \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) is not an identity. It only holds when \( \sin \theta \) is non-negative, which isn't true for all \( \theta \).

Key Concepts

Pythagorean Identity

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables. One of the fundamental identities is the Pythagorean Identity:

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

This identity is essential in trigonometry because it defines a relationship between the sine and cosine of an angle. By rearranging this identity, we can express

• \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \)
• \( \cos^2 \theta \) as \( 1 - \sin^2 \theta \).

These expressions are particularly useful when solving problems involving trigonometric functions.
In the context of our original exercise, the identity allows us to transform and analyze or verify other equations, showing whether they hold true over their range. It's a powerful tool for proving other identities or simplifying expressions. Understanding and remembering the Pythagorean Identity is crucial for anyone studying trigonometry.

Quadrants of the Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It plays a vital role in understanding trigonometric functions and their properties. The circle is divided into four quadrants, each with unique properties that influence the sign of trigonometric functions:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, but cosine is positive.

Each quadrant corresponds to a specific range of angle measurements: 0 to 90 degrees, 90 to 180 degrees, 180 to 270 degrees, and 270 to 360 degrees respectively.
This understanding helps explain why \( \sin \theta \) evaluates differently depending on the angle's location on the circle. So, when trying to determine if an equation is an identity, it is essential to consider the quadrant, since the sign of the trigonometric functions can change.

Positive and Negative Values of Sine

The sine function, often written as \( \sin \theta \), represents the vertical component of the radius of the unit circle. As you move through different quadrants, the sign of \( \sin \theta \) changes following the properties of these quadrants:

• In Quadrant I (0 to 90 degrees), sine values are positive.
• In Quadrant II (90 to 180 degrees), sine values remain positive.
• In Quadrant III (180 to 270 degrees), sine values become negative.
• In Quadrant IV (270 to 360 degrees), sine values are negative again.

When an equation like \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) is presented, it assumes sine is only positive, due to the square root operation. However, as seen, sine doesn't always hold a positive value, which depends on the angle's quadrant.
This understanding is crucial because it shows the importance of correcting the equation to \( \sin \theta = \pm \sqrt{1 - \cos^2 \theta} \) to accurately reflect that sine can be both positive and negative, confirming the equation's identity status.