Problem 2 - Algebra 2 and Trigonometry

Question

For what values of \(\theta\) is \(\sin \theta=\sqrt{1-\cos ^{2} \theta}\) true?

Step-by-Step Solution

Answer

The expression is always true for positive values of \(\sin \theta\).

Step 1: Recognize the Identity

Notice that \(\sin^2 \theta = 1 - \cos^2 \theta\) is a known trigonometric identity. This implies that \(\sin \theta = \sqrt{1-\cos^2 \theta}\) is always true for \(\sin \theta\).

Step 2: Consider the Positive Root

Since \(\sin \theta = \sqrt{1-\cos^2 \theta}\) by definition, the positive square root corresponds directly to the sine values in the first quadrant and possibly the second quadrant, where sine is positive.

Step 3: Consider All Possible Values

The expression \(\sin \theta = \sqrt{1-\cos^2 \theta}\) holds true for all \(\theta\) where \(\sin \theta\) is positive, meaning \(\theta\) can be any angle where the sine value is positive.

Key Concepts

Understanding the Sine Function

The sine function, often denoted as \( \sin \theta \), is a fundamental component of trigonometry. It originates from the unit circle, where it represents the y-coordinate of a point on the circle's circumference corresponding to an angle \( \theta \). This function is periodic with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) radians.

Key characteristics of the sine function include:

• The range: \([-1, 1]\), meaning sine values cannot exceed 1 or be less than -1.
• It is an odd function, which means that \( \sin(-\theta) = -\sin \theta \).
• It has zeros at integer multiples of \( \pi \).

Understanding the geometric interpretation, sine measures the vertical distance (or height) from the x-axis to the line segment representing the terminal side of the angle \( \theta \) on the unit circle. This concept is crucial when solving problems involving right triangles as it relates directly to the opposite side over the hypotenuse ratio.

Exploring the Cosine Function

The cosine function, denoted as \( \cos \theta \), is equally important in trigonometry. It pertains to the x-coordinate of a point on the unit circle that matches the angle \( \theta \). Similar to the sine function, it also has a period of \( 2\pi \).

The cosine function's features include:

• The range: \([-1, 1]\).
• Zero crossings at odd multiples of \( \pi/2 \).
• It is an even function, meaning \( \cos(-\theta) = \cos \theta \).

In terms of trigonometric identities, the cosine function often pairs with the sine function. One crucial relationship is the Pythagorean identity: \( \cos^2 \theta + \sin^2 \theta = 1 \). In geometric terms, cosine measures the horizontal distance from the origin to the terminal side of the angle in the unit circle. This relationship is vital when relating angles to x-coordinates in the coordinate system.

Understanding Positive Angles

In trigonometry, angles are typically measured in standard position, starting from the positive x-axis and moving counterclockwise. These are what we refer to as positive angles.

When talking about angles where the sine function is positive, we usually refer to angles in the first and second quadrants.

• In the first quadrant, both sine and cosine are positive.
• In the second quadrant, sine remains positive while cosine becomes negative.

Recognizing these properties is essential to solve equations where a function, like sine, is equated to a positive value, such as \( \sqrt{1 - \cos^2 \theta} \). In this scenario, the condition of the square root being positive aligns with sine being positive. Therefore, for angles in either the first or second quadrant, \( \sin \theta = \sqrt{1 - \cos^2 \theta} \) is valid.

This understanding helps in identifying the correct subset of angles where this trigonometric identity holds true, ensuring accurate solutions to problems.