Problem 6 - 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. \(\tan \theta\) \(\left(-\frac{1}{4}, y\right),\) third quadrant

Step-by-Step Solution

Answer

a. \(-\frac{\sqrt{15}}{4}\); b. \(-\frac{1}{4}\); c. \(-\frac{\sqrt{15}}{4}\); d. \(\sqrt{15}\).

Step 1: Determine the y-coordinate from the unit circle equation

Since the point \( P \) is on the unit circle, the equation \( x^2 + y^2 = 1 \) holds true. Given that \( x = -\frac{1}{4} \), substitute this into the equation to find \( y \).\[ (-\frac{1}{4})^2 + y^2 = 1 \]\[ \frac{1}{16} + y^2 = 1 \]\[ y^2 = 1 - \frac{1}{16} \]\[ y^2 = \frac{16}{16} - \frac{1}{16} \]\[ y^2 = \frac{15}{16} \]Since we are in the third quadrant where \( y < 0 \), we take the negative root: \[ y = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4} \].

Step 2: Calculate \(\cos \theta\)

In the unit circle, the \(\cos \theta\) corresponds to the \(x\)-coordinate of the point \(P\). Therefore, \(\cos \theta = x = -\frac{1}{4} \).

Step 3: Calculate \(\sin \theta\)

Similarly, \(\sin \theta\) corresponds to the \(y\)-coordinate of the point \(P\). Therefore, \(\sin \theta = y = -\frac{\sqrt{15}}{4} \).

Step 4: Calculate \(\tan \theta\)

The tangent function is defined as the ratio of the sine to the cosine. So, \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{15}}{4}}{-\frac{1}{4}} = \sqrt{15} \].

Key Concepts

Trigonometric Functions

Trigonometric functions help us make sense of relationships within triangles, particularly right-angled triangles. These functions include sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. These are defined as:

• Sine ( \( ext{sin} \)): This function gives the ratio of the length of the side of the triangle opposite the angle to the hypotenuse.
• Cosine ( \( ext{cos} \)): This function gives the ratio of the adjacent side to the hypotenuse.
• Tangent ( \( ext{tan} \)): This function is the ratio of the opposite side to the adjacent side.

In the context of a unit circle, where the circle's radius is 1, these functions can be directly understood from the coordinates of a point on the circle. So, if you have an angle \( heta \) with coordinates on the unit circle \((x, y) \), then:

• \( ext{cos} heta = x \)
• \( ext{sin} heta = y \)
• \( ext{tan} heta = rac{y}{x} \)

Standard Position

An angle is in "standard position" if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. From this starting point, the angle's terminal side rotates around the origin. Measurements of angles in standard position are typically in radians or degrees and are used to find the angle's coordinates on the unit circle. The direction of rotation determines the angle's sign:

• Counter-clockwise rotation: Positive angle
• Clockwise rotation: Negative angle

Working with angles in standard position helps in understanding Trigonometric Function values at various points on the unit circle. It becomes especially relevant when discussing angles that intersect the unit circle, as the point where the terminal side of the angle intersects gives us the relevant sine and cosine values.

Quadrants

The coordinate plane is divided into four sections, known as quadrants, which help us determine the signs of trigonometric functions. They are:

• First Quadrant: Both sine and cosine values are positive because both x and y coordinates are positive.
• Second Quadrant: Sine values are positive, but cosine values are negative as the x coordinate is negative.
• Third Quadrant: Both sine and cosine are negative. Here, both coordinates (x, y) are negative. This is why in the original exercise, it was concluded that \( y = -\frac{\sqrt{15}}{4} \).
• Fourth Quadrant: Cosine values are positive, while sine values are negative, since the y coordinate is negative.

Understanding these quadrants is crucial as each quadrant affects the sign of the trigonometric functions and subsequently affects equations and calculations involving these functions.

Sine and Cosine

Sine and cosine are foundational trigonometric functions that relate angles in the unit circle to coordinates. This is incredibly useful for solving problems like the one in the original exercise, where knowing the x or y coordinate allows you to determine the other.

• Sine ( \( ext{sin} \)): Represents the y-coordinate of a point on the unit circle. It reveals how far 'up' or 'down' the point is from the x-axis.
• Cosine ( \( ext{cos} \)): Represents the x-coordinate of a point on the unit circle. It shows the distance 'across' from the origin.

In the unit circle, because the radius is 1, \( ext{sin}^2 heta + ext{cos}^2 heta = 1 \), an equation derived from Pythagorean identity, holds true for any angle \( heta \). This allows us to calculate one if we know the other. In problems involving quadrants, it's important to remember that signs of these functions change depending on which quadrant the terminal side of the angle lies.