Problem 75 - Precalculus: Functions and Graphs

Question

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. II; on the line \(y=-4 x\)

Step-by-Step Solution

Answer

The trigonometric functions are: \(\sin(\theta)=\frac{4}{\sqrt{17}}\), \(\cos(\theta)=\frac{-1}{\sqrt{17}}\), \(\tan(\theta)=-4\), \(\csc(\theta)=\frac{\sqrt{17}}{4}\), \(\sec(\theta)=-\sqrt{17}\), \(\cot(\theta)=-\frac{1}{4}\).

Step 1: Understand the Problem

The problem provides that the angle \(\theta\) is in the second quadrant and on the line \(y = -4x\). In the second quadrant, \(x\) is negative and \(y\) is positive.

Step 2: Establish a Point on the Line

Since the angle is in the second quadrant and the line equation is \(y = -4x\), we can find a point by choosing \(x = -1\). This gives \(y = -4(-1) = 4\). Hence, the point \((-1, 4)\) lies on the line and in the correct quadrant.

Step 3: Calculate the Radius r

The radius \(r\) is the distance from the origin to the point \((-1, 4)\). It can be calculated using the distance formula: \(r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}\).

Step 4: Compute the Trigonometric Functions

Using the point \((-1, 4)\) and the radius \(r = \sqrt{17}\), calculate the six trigonometric functions.- \(\sin(\theta) = \frac{y}{r} = \frac{4}{\sqrt{17}}\)- \(\cos(\theta) = \frac{x}{r} = \frac{-1}{\sqrt{17}}\)- \(\tan(\theta) = \frac{y}{x} = \frac{4}{-1} = -4\)- \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\sqrt{17}}{4}\)- \(\sec(\theta) = \frac{1}{\cos(\theta)} = -\sqrt{17}\)- \(\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{1}{4}\)

Key Concepts

second quadrant

When dealing with trigonometric functions and angles in the coordinate plane, it's important to understand the concept of quadrants. The coordinate plane is divided into four quadrants:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

The second quadrant is particularly interesting for trigonometric functions because it affects the signs of sine and cosine. In the second quadrant:

• Sin(θ) is positive, because the y-coordinate is positive, which represents the opposite side of a right triangle.
• Cos(θ) is negative, due to the negative x-coordinates in this section of the plane, affecting the adjacent side of the triangle.
• Tangent (Tan(θ)) is negative because it is the ratio of sine to cosine (positive divided by negative).

The understanding of these signs is crucial when solving problems involving trigonometric functions in different quadrants.

line equation

The equation of a line in the two-dimensional plane describes all points that lie on that line. A common form of a line equation is given by the slope-intercept form, which is:\[ y = mx + b \]where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

For the line \(y = -4x\), there is no y-intercept \(b\), meaning it passes through the origin (0,0). The slope here is -4. This tells us that for any horizontal move of 1 unit to the right, the line goes down 4 units. In the context of the original problem, choosing a negative x-value, such as -1, ensures that the point (-1, 4) lies within the second quadrant, complying with the given conditions of the exercise.

distance formula

The distance formula is an essential tool in coordinate geometry for calculating the distance between two points in the plane. It is derived from the Pythagorean theorem and is expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This measures how far apart the points are. In trigonometry, one of these points is often the origin (0, 0), simplifying the formula to:\[ r = \sqrt{x^2 + y^2} \]where \(r\) is known as the radius or hypotenuse in the context of a right-angled triangle formed by the x-axis, y-axis, and the line through a particular coordinate. For the point \((-1, 4)\), the distance, or radius \(r\), is calculated as \(\sqrt{(-1)^2 + 4^2} = \sqrt{17}\). This radius is crucial for determining the exact values of the trigonometric functions.

standard position

In trigonometry, an angle is said to be in standard position if it is measured from the positive x-axis, moving counterclockwise. This standardization allows for a uniform way to define and use angles on the coordinate plane. When angles are described this way, their terminal sides determine which quadrant they fall into.Understanding standard position is crucial because it sets the framework for determining the sine and cosine of angles based on their coordinates:

• The initial side of the angle lies along the positive x-axis.
• The terminal side situated in different quadrants influences the angle's sine and cosine values.

When an angle's terminal side is on a line, such as \(y = -4x\), it's useful to translate points from this line into trigonometric functions. This approach simplifies calculations and helps visualize angles and their respective functions better.