Problem 12 - 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. \(\tan \theta\) \(\left(-\frac{\sqrt{7}}{3}, y\right),\) second quadrant

Step-by-Step Solution

Answer

a) \(\frac{\sqrt{2}}{3}\), b) \(-\frac{\sqrt{7}}{3}\), c) \(\frac{\sqrt{2}}{3}\), d) \(-\frac{\sqrt{2}}{\sqrt{7}}\).

Step 1: Use the Pythagorean Identity for the Unit Circle

Since the point \(P\) is on the unit circle, we use the identity \(x^2 + y^2 = 1\). Substitute \(x = -\frac{\sqrt{7}}{3}\) into the equation: \((-\frac{\sqrt{7}}{3})^2 + y^2 = 1\).

Step 2: Solve for the y-coordinate

Calculate \((-\frac{\sqrt{7}}{3})^2 = \frac{7}{9}\). Substitute it back: \(\frac{7}{9} + y^2 = 1\). Solve for \(y^2\): \(y^2 = 1 - \frac{7}{9} = \frac{2}{9}\). Thus, \(y = \pm\frac{\sqrt{2}}{3}\). Because the point is in the second quadrant, \(y\) must be positive. Therefore, \(y = \frac{\sqrt{2}}{3}\).

Step 3: Determine \(\cos \theta\)

\(\cos \theta\) is the \(x\)-coordinate of point \(P\) which is \(-\frac{\sqrt{7}}{3}\).

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

\(\sin \theta\) is the \(y\)-coordinate of point \(P\) which we found to be \(\frac{\sqrt{2}}{3}\).

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

\(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\). So, \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{2}}{3}}{-\frac{\sqrt{7}}{3}} = -\frac{\sqrt{2}}{\sqrt{7}}\).

Key Concepts

Pythagorean Identity

One of the most fundamental concepts used in trigonometry is the Pythagorean Identity, especially when dealing with the unit circle. It states that for any angle \( \theta \) with point \( P(x, y) \) on the unit circle, \( x^2 + y^2 = 1 \). This formula emphasizes the relationship between the x- and y-coordinates of points lying on the circle and is derived from the Pythagorean theorem.
For the problem at hand, the Pythagorean Identity helps us find the missing coordinate once we know one coordinate. By substituting the given x-coordinate of \( -\frac{\sqrt{7}}{3} \) into the equation, we solve for \( y \), yielding two possible values. However, knowing the quadrant (second quadrant in this case) dictates which value to choose. In the second quadrant, the y-coordinate must be positive, informing us to select \( y = \frac{\sqrt{2}}{3} \). This step is crucial to ensure valid and reliable trigonometric calculations.

Trigonometric Ratios

Trigonometric ratios provide a way to relate the angle \( \theta \) with the coordinates of point \( P \) on a unit circle. These ratios are:

• Cosine (cos): This is the x-coordinate of the point P. For this exercise, \( \cos \theta = -\frac{\sqrt{7}}{3} \).
• Sine (sin): This represents the y-coordinate, which we found to be \( \sin \theta = \frac{\sqrt{2}}{3} \).
• Tangent (tan): It is the ratio of the sine to the cosine, expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{\sqrt{2}}{\sqrt{7}} \).

These ratios are fundamental in connecting the unit circle to the angle’s position and assisting in solving trigonometric problems. Notice how knowing one trigonometric ratio can lead us to find others, as demonstrated by finding \( \tan \theta \) after determining \( \sin \theta \) and \( \cos \theta \).
Using these ratios helps in visualizing and analyzing the behavior of functions at different angles, enabling us to interpret and solve diverse trigonometric scenarios.

Quadrants

The unit circle is divided into four quadrants, each representing a different combination of signs for the x- and y-coordinates of a point. These quadrants are crucial for determining the signs of trigonometric functions:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

In our exercise, point \( P \) lies in the second quadrant, suggesting that any trigonometric function related specifically to y will be positive, while those based on x will be negative. This quadrant information not only helps verify coordinate values but also directly affects the evaluation of trigonometric functions, ensuring they are correct.
Quadrant determination is a simple yet powerful tool in confirming the reliability and accuracy of trigonometric solutions, ensuring they correspond to the correct angles and positions on the unit circle.