Problem 5 - 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. \(\tan \theta\) \(\left(-\frac{6}{10}, y\right),\) second quadrant

Step-by-Step Solution

Answer

a. \(y = 0.8\), b. \(\cos \theta = -0.6\), c. \(\sin \theta = 0.8\), d. \(\tan \theta = -\frac{4}{3}\)

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Any point \((x, y)\) on this circle will satisfy the equation \(x^2 + y^2 = 1\). The coordinates represent values of trigonometric functions for an angle \(\theta\).

Step 2: Solve for the y-coordinate

Given the x-coordinate \(x = -\frac{6}{10} = -0.6\), substitute this value into the unit circle equation: \(x^2 + y^2 = 1\). Thus, \((-0.6)^2 + y^2 = 1\). Simplify to find \(y^2 = 1 - 0.36 = 0.64\), so \(y = \pm 0.8\). Since the point is in the second quadrant where y is positive, \(y = 0.8\).

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

For a point \((x, y)\) on the unit circle, \(\cos \theta\) is the x-coordinate. Thus, \(\cos \theta = -0.6\).

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

For a point \((x, y)\) on the unit circle, \(\sin \theta\) is the y-coordinate. Thus, \(\sin \theta = 0.8\).

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

The tangent function is determined by the ratio \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute in the values found: \(\tan \theta = \frac{0.8}{-0.6} = -\frac{4}{3}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. These functions are essential in analyzing and understanding angles and their measurement in various fields.
In the context of a unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane, trigonometric functions provide valuable insights into identifying the sine, cosine, and tangent of any given angle \(\theta\).

• Trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).
• They portray how the triangle's sides compare to the angles measured around a circle.
• These functions help navigate through different quadrants of a unit circle, each of which represents different \(x\) and \(y\) sign conventions.

Understanding these relationships is key when solving problems that involve finding unknown side lengths or angles in right triangles.

Cosine Function

The cosine function, shortened as \(\cos\), presents the ratio between the adjacent side over the hypotenuse in a right triangle. On the unit circle, the cosine of an angle is represented by the \(x\)-coordinate of a point.
When the angle \(\theta\) is positioned in a standard position, it starts from the positive x-axis. The terminal side of the angle will intersect the unit circle:

• In our exercise, the given x-coordinate is \(-0.6\)
• This directly gives us \(\cos \theta = -0.6\)
• The sign of the cosine depends on which quadrant the angle is in (2nd quadrant in this case)

It's important to remember that on a unit circle, \(x^2 + y^2 = 1\) remains true since the radius is always 1.

Sine Function

The sine function, commonly described as \(\sin\), is the ratio of the opposite side to the hypotenuse in a right triangle. In the framework of the unit circle, the sine of an angle is equivalent to the \(y\)-coordinate of a point.
The sine function provides clarity on the vertical position of a point as it moves around the circle:

• In our scenario, the exercise gives the x-coordinate as \(-0.6\)
• We calculated \(y = 0.8\) using the equation \(x^2 + y^2 = 1\)
• Thus, \(\sin \theta = 0.8\)
• The positive sign in the 2nd quadrant confirms that \(y\) is positive.

The sine function helps in determining how much a point on the unit circle is displaced vertically from the center.

Tangent Function

The tangent function, expressed as \(\tan\), is the ratio of the sine to the cosine of an angle. It provides insights into the slope of the line formed by the terminal side and the x-axis. This information is valuable when you need to understand the relationship between the vertical and horizontal components of an angle:
For any angle \(\theta\), the tangent function \(\tan \theta\) on the unit circle is found using:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• Substitute \(\sin \theta = 0.8\) and \(\cos \theta = -0.6\), yielding \(\tan \theta = -\frac{4}{3}\)
• The negative sign indicates the direction of the function in our specific quadrant.

Understanding the tangent function is crucial when dealing with slope calculations or any applications involving steepness and inclination.