Problem 2 - Algebra 2 and Trigonometry

Question

Hannah said that if \(\cos \theta=a,\) then \(\sin \theta=\pm \sqrt{1-a^{2}} .\) Do you agree with Hannah? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Hannah is correct because \\( \sin \theta = \pm \sqrt{1-a^2} \\) depending on the quadrant of \\( \theta \\).

Step 1: Understanding Basic Trigonometric Identity

The Pythagorean identity states that \(\cos^2 \theta + \sin^2 \theta = 1\). This identity is fundamental for relating the sine and cosine of the same angle.

Step 2: Expressing Sine in Terms of Cosine

Given \( \cos \theta = a \), we substitute \( a \) into the Pythagorean identity: \( a^2 + \sin^2 \theta = 1 \). This gives us \(|\sin \theta|= \sqrt{1-a^2} \).

Step 3: Considering the Quadrants

Since \( |\sin \theta| = \sqrt{1-a^2} \), the \( \sin \theta \) could indeed be \( \pm \sqrt{1-a^2} \), depending on which quadrant \( \theta \) is in. If \( \theta \) is in the first or second quadrant, \( \sin \theta \) is positive; if it is in the third or fourth quadrant, \( \sin \theta \) is negative.

Key Concepts

Trigonometric Identities

Trigonometric identities are like the glue that holds various trigonometric concepts together. These identities help us relate different trigonometric functions and solve equations involving them. A key identity to remember is the Pythagorean identity:

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

This identity tells us something important: The square of the cosine of an angle plus the square of the sine of the same angle always equals one. This is due to the geometric properties of the unit circle. In the context of our problem, if we know \( \cos \theta = a \), we can substitute this into our identity and solve for \( \sin \theta \) to get \( \sin^2 \theta = 1-a^2 \). Consequently, \( \sin \theta \) becomes \( \pm \sqrt{1-a^2} \). The plus or minus depends on the angle's position on the coordinate plane, which leads us to our next concept.

Sine and Cosine Relationship

The relationship between sine and cosine is deeply rooted in their definition on the unit circle. On the unit circle, the cosine of an angle represents the x-coordinate, while the sine represents the y-coordinate. These are linked through the identity \( \cos^2 \theta + \sin^2 \theta = 1 \). Let's break it down further:

• When you know the cosine of an angle, you can find the sine by rearranging the Pythagorean identity.
• If \( \cos \theta = a \), substitute into \( a^2 + \sin^2 \theta = 1 \), yielding \( \sin^2 \theta = 1-a^2 \).
• Thus, \( \sin \theta = \pm \sqrt{1-a^2} \).

The relationship is symmetrical, stating that if you know one, you can find the other. This reveals the inherent link between these two functions and shows why understanding one helps to understand the other.

Angle Quadrants

To determine the sign of \( \sin \theta \), we need to consider the angle's quadrant. The unit circle is divided into four quadrants, each affecting the signs of sine and cosine:

• 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.

For our exercise, knowing \( \cos \theta = a \) leaves us with two potential values for \( \sin \theta \): \( +\sqrt{1-a^2} \) or \( -\sqrt{1-a^2} \). To choose correctly, identify which quadrant \( \theta \) lies in. For example, if \( \theta \) is in the second quadrant, \( \sin \theta \) will be positive. This classification means the angle’s position on the circle directly influences which sign is correct for sine in the expression \( \pm \sqrt{1-a^2} \).