Problem 4 - 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. \(\tan \theta\) \(\left(\frac{5}{13}, y\right),\) fourth quadrant

Step-by-Step Solution

Answer

a. \( y = -\frac{12}{13} \); b. \( \cos \theta = \frac{5}{13} \); c. \( \sin \theta = -\frac{12}{13} \); d. \( \tan \theta = -\frac{12}{5} \).

Step 1: Understand the Unit Circle

In a unit circle, any point \(P\) on the circle can be expressed in terms of \(x\) and \(y\) coordinates, where \(x^2 + y^2 = 1\). Since \(x\) is given as \(\frac{5}{13}\), use this relationship to find \(y\).

Step 2: Calculate the y-coordinate

Using the equation for a unit circle \(x^2 + y^2 = 1\), substitute \(x = \frac{5}{13}\):\[ \left(\frac{5}{13}\right)^2 + y^2 = 1 \rightarrow \frac{25}{169} + y^2 = 1\]. Solve for \(y^2\):\[ y^2 = 1 - \frac{25}{169} = \frac{169}{169} - \frac{25}{169} = \frac{144}{169} \] Taking the square root of both sides gives \( y = \pm \frac{12}{13} \). As the point is in the fourth quadrant, where \(y\) is negative, \( y = -\frac{12}{13} \).

Step 3: Confirm the Trigonometric Functions

Since \(P\) is on a unit circle, the coordinates \((x, y)\) = \( \left(\frac{5}{13}, -\frac{12}{13}\right)\), equates to \(\cos \theta = \frac{5}{13}\) and \(\sin \theta = -\frac{12}{13}\).

Step 4: Calculate the tan(θ)

Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the known values gives:\[ \tan \theta = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5} \]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics that help us understand relationships in a unit circle. The primary trigonometric functions are:

• Sine (): This is defined as the y-coordinate of a point on the unit circle. It tells us how far up or down the point lies from the x-axis.
• Cosine (): This is the x-coordinate of a point on the unit circle. It refers to how far left or right the point lies from the y-axis.
• Tangent (): This ratio,  / , gives the slope of the terminal side of the angle relative to the x-axis. It's helpful in determining the steepness.

In this specific problem, the unit circle helps in expressing trigonometric identities. For example, knowing that  gives us the y-value, and  gives us the x-coordinate, we can unearth other functions like  with these properties. Understanding these functions provides a foundation for deeper studies in mathematics, like calculus or engineering.

Angles in Standard Position

Angles in standard position are measured starting from the positive x-axis and moving in the direction specified by the rotation. This concept helps in determining the location of various trigonometric functions on the unit circle.

To identify angles in standard position, remember:

• The vertex of the angle is positioned at the origin of a coordinate plane.
• Its initial side is always on the positive x-axis.
• The movement of the terminal side defines its rotation. Counter-clockwise rotations are considered positive while clockwise rotations are negative.

Once an angle is placed in this standard position, finding its related trigonometric functions becomes more straightforward, especially within the context of the unit circle. An understanding of standard positions helps us visually grasp where each trigonometric function lies, making math feel less abstract and more grounded in geometry.

Fourth Quadrant

The fourth quadrant, found in the lower right section of a Cartesian plane, holds unique properties that influence trigonometric calculations. Understanding these properties is crucial since it affects the signs and values of trigonometric functions.

• Coordinates: In the fourth quadrant, x-values are positive while y-values are negative.
• Function Signs: Here,  (y), is negative, while  (x), is positive. This makes  always negative and  positive.
• Angles: Any angle whose terminal side lies in this quadrant reflects a clockwise rotation from the positive x-axis.

When solving problems involving the fourth quadrant, it's beneficial to consider these properties. They guide the determination of function values and prevent common sign errors, helping you solve problems with greater confidence.