Problem 34 - Precalculus: A Concise Course

Question

The terminal side of \(\boldsymbol{\theta}\) lies on the given line in the specified quadrant. Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) by finding a point on the line. $$ y=\frac{1}{3} x $$, III

Step-by-Step Solution

Answer

The six trigonometric functions for the angle \(\theta\) are: \(\sin(\theta)=-\frac{1}{\sqrt{10}}\), \(\cos(\theta)=-\frac{3}{\sqrt{10}}\), \(\tan(\theta)=\frac{1}{3}\), \(\csc(\theta)=-\sqrt{10}\), \(\sec(\theta)=-\frac{\sqrt{10}}{3}\), and \(\cot(\theta)=-3\).

Step 1: Finding a point on the line

The line equation \(y=\frac{1}{3}x\) suggests that the gradient is 1/3. This means, for example, when \(x=-3\), \(y=-1\). These values are chosen in this way because we are dealing with quadrant III, where both \(x\) and \(y\) are negative. Thus, a point on the line within quadrant III would be (-3, -1).

Step 2: Determining the values for x, y, and r

From the point in step 1, we get \(x=-3\) and \(y=-1\). The hypotenuse \(r\) represents the distance from the origin (0,0) to this point. We can calculate it using the Pythagorean theorem, \(r=\sqrt{x^2+y^2}=\sqrt{(-3)^2+(-1)^2}=\sqrt{10}\). Now, we have values for \(x\), \(y\), and \(r\). Note: In quadrant III, \(r\) is considered positive.

Step 3: Calculating the six trigonometric functions

With the values obtained in step 2, it's possible to calculate each trigonometric function for \(\theta\) as such: \(\sin(\theta)=\frac{y}{r}=-\frac{1}{\sqrt{10}}\), \(\cos(\theta)=\frac{x}{r}=-\frac{3}{\sqrt{10}}\), \(\tan(\theta)=\frac{y}{x}=\frac{1}{3}\), \(\csc(\theta)=\frac{r}{y}=-\sqrt{10}\), \(\sec(\theta)=\frac{r}{x}=-\frac{\sqrt{10}}{3}\), \(\cot(\theta)=\frac{x}{y}=-3\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system used to specify the position of a point in a plane by using two numbers, which are usually denoted as \(x\) and \(y\). This mathematical concept helps locate any point in a two-dimensional space. The \(x\)-coordinate shows how far to move horizontally from the origin, while the \(y\)-coordinate indicates the vertical movement.

• The origin, represented as (0,0), is the central point where both axes intersect.
• The horizontal axis is known as the x-axis, and the vertical is the y-axis.
• The coordinates (x,y) establish a point's position relative to the origin.

Each point on the plane is represented uniquely by these two numbers, making Cartesian coordinates a fundamental concept in geometry and trigonometry.

Quadrant III

The Cartesian plane is divided into four quadrants. Each quadrant is defined by the signs of the coordinates of the points located within it. In Quadrant III, both the x and y coordinates are negative. This is a key detail when analyzing points and their positions, especially for understanding trigonometric functions like sine and cosine.

• Quadrant I: Positive x, positive y
• Quadrant II: Negative x, positive y
• Quadrant III: Negative x, negative y
• Quadrant IV: Positive x, negative y

When working within Quadrant III, one must always remember that x and y values are negative, which directly affects the computation of trigonometric functions and other mathematical operations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics used to find the length of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is often expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

• Useful in determining the distance between two points in a plane.
• In our exercise: \(a = -3\), \(b = -1\), and \(c=sqrt{10}\).
• Application in calculating the hypotenuse \(r\) for the coordinates (-3,-1).

This theorem not only serves triangles but is a powerful tool in trigonometry when calculating the trigonometric functions of angles.

Line Equation

A line equation in the form \(y=mx+b\) describes a straight line where \(m\) is the slope (gradient), and \(b\) is the y-intercept. The slope, \(m\), represents how steep the line is. In the given problem, the line equation is \(y=\frac{1}{3}x\), indicating a slope of \(\frac{1}{3}\) and no vertical shift, meaning the line crosses the origin.

• The slope \(m\) determines the line's angle concerning the x-axis.
• In simpler terms, for every 3 units moved along the x-axis, the line moves 1 unit up the y-axis.
• Understanding the line's position helps find specific points required for trigonometric evaluations.

Grasping how to interpret and use a line equation aids in solving various mathematical and real-world problems involving linear relationships.