Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-11, P\) is the point at which the terminal side of an angle in standard position intersects the unit circle. The measure of the angle is \(\theta .\) For each point \(P\) the \(x\) -coordinate and the quadrant is given. Find: a. the \(y\) -coordinate of \(P\) b. \(\cos \theta\) c. \(\sin \theta\) d. \(\tan \theta\) \(\left(-\frac{\sqrt{2}}{2}, y\right),\) third quadrant

Step-by-Step Solution

Answer

The coordinates are \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\), \( \cos \theta = -\frac{\sqrt{2}}{2} \), \( \sin \theta = -\frac{\sqrt{2}}{2} \), \( \tan \theta = 1 \).

Step 1: Identify Coordinates

For a point located on the unit circle, the coordinates are represented as \((x, y)\). Here, the given \(x\)-coordinate is \(-\frac{\sqrt{2}}{2}\), and the point is in the third quadrant. We need find the corresponding \(y\)-coordinate, noting the properties of the unit circle.

Step 2: Use the Pythagorean Identity

The equation for a point \((x, y)\) on the unit circle is \(x^2 + y^2 = 1\). Substitute the given \(x\)-coordinate into the equation: \[ \left(-\frac{\sqrt{2}}{2}\right)^2 + y^2 = 1 \] Which simplifies to \[ \frac{2}{4} + y^2 = 1 \] or \[ \frac{1}{2} + y^2 = 1 \] and solve for \(y^2\).

Step 3: Solve for y

Rearrange the equation: \[ y^2 = 1 - \frac{1}{2} \] \[ y^2 = \frac{1}{2} \] Therefore, \(y\) can be positive or negative square root of \(\frac{1}{2}\). However, because the point is in the third quadrant, where the \(y\)-coordinate is negative, we have: \[ y = -\frac{\sqrt{2}}{2} \]

Step 4: Determine \(\cos \theta\) and \(\sin \theta\)

For angles in standard position on the unit circle, \(\cos \theta\) is given by the \(x\)-coordinate, and \(\sin \theta\) is given by the \(y\)-coordinate. Thus: \[ \cos \theta = -\frac{\sqrt{2}}{2} \] \[ \sin \theta = -\frac{\sqrt{2}}{2} \]

Step 5: Compute \(\tan \theta\)

\(\tan \theta\) is given by the ratio \(\frac{\sin \theta}{\cos \theta}\). Substitute the values from previous steps: \[ \tan \theta = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \]

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate an angle of a right triangle to ratios of two side lengths. These functions are vital in understanding the properties and behaviors of angles, especially when working with the unit circle. On a unit circle, which has a radius of one unit, points are expressed in the form \((x, y)\), where:

• \(\cos \theta = x\)
• \(\sin \theta = y\)
• \(\tan \theta = \frac{y}{x}\)

Each of these functions serves a purpose in determining the relationships between angles and distances in a circular motion.
For example, if you know the \(x\)-coordinate of a point on the unit circle, you can easily find \(\cos \theta\) because they are the same. Similarly, you can find \(\sin \theta\) using the \(y\)-coordinate. Trigonometric functions are also applicable beyond angles, extending to wave functions and harmonic motion in physics.

Pythagorean Identity

The Pythagorean Identity is a fundamental relation in trigonometry, named after the famous Pythagorean Theorem. It states that for any angle \(\theta\) located on the unit circle, the equation is:

• \(x^2 + y^2 = 1\)

This identity arises because the unit circle is derived from a right triangle, where the hypotenuse is always 1.
Therefore, the squares of the sine and cosine at any angle will equal 1. In practical terms, this equation allows us to find the missing coordinate if one of \(x\) or \(y\) is known.
For example, if \(x = -\frac{\sqrt{2}}{2}\), substituting that into the identity, you can solve for \(y\) using:\[(-\frac{\sqrt{2}}{2})^2 + y^2 = 1\]Applying the identity helps solve many problems involving the unit circle and angles lying on it.

Standard Position

When an angle is in its "standard position," its vertex is at the origin of the coordinate plane, and its initial side lies along the positive \(x\)-axis. The angle then sweeps counterclockwise to reach its terminal side, the line where it stops.
This concept is used to measure angles consistently as either positive or negative based on their direction. Positive angles move counterclockwise, while negative angles move clockwise.
In the unit circle, the terminal side often intersects at specific key points, which helps in accurately using trigonometric functions and identities.
For example, if \(P(-\frac{\sqrt{2}}{2}, y)\) is a point of intersection in standard position, then \(\theta\) represents the degree measure from the positive \(x\)-axis to that point.

Third Quadrant

The coordinate plane is divided into four quadrants, and the third quadrant is located in the bottom left, characterized by both \(x\) and \(y\) coordinates being negative. It spans angles from 180 to 270 degrees.
This quadrant is important in trigonometry because it affects the positivity or negativity of trigonometric functions. For any angle in the third quadrant:

• \(\cos \theta < 0\)
• \(\sin \theta < 0\)
• \(\tan \theta > 0\)

The positive value of \(\tan \theta\) arises because both sine and cosine are negative, and their ratio results in a positive tangent. Recognizing which quadrant the angle resides in is essential because it influences the sign and properties of the trigonometric calculations.
Understanding these concepts help when solving problems related to the unit circle and angles, especially when deriving the exact values for these functions.