Problem 40 - A Graphical Approach to Precalculus with Limits

Question

An equation of the terminal side of an angle \(\theta\) in standard position is given with a restriction on \(x\). Sketch the least positive angle \(\theta\), and find the values of the six trigonometric functions of \(\theta\). $$-5 x-3 y=0, x \leq 0$$

Step-by-Step Solution

Answer

\(\theta = 121^\circ\); \( \sin(\theta) = \frac{5}{\sqrt{34}}, \cos(\theta) = -\frac{3}{\sqrt{34}}, \tan(\theta) = -\frac{5}{3}\).

Step 1: Transform the Given Equation

First, we need to find the slope-intercept form of the given equation \(-5x - 3y = 0\). To do that, solve for \(y\): \[-3y = 5x\] \[y = -\frac{5}{3}x\].Plotting this, we have a line with a slope of \(-\frac{5}{3}\), which passes through the origin \((0,0)\). Since \(x \leq 0\), the terminal side lies in the second quadrant of the coordinate plane.

Step 2: Identify the Angle and Terminal Side

In standard position, the angle \(\theta\) is measured from the positive \(x\)-axis to the terminal side, counter-clockwise. The line \(y = -\frac{5}{3}x\) with \(x \leq 0\) indicates an angle in the second quadrant. Since the line has a negative slope, the terminal side moves downwards to the left, showing the line forms an obtuse angle with the positive \(x\)-axis.

Step 3: Calculate the Angle of the Line

We find the reference angle \(\alpha\) by calculating the arctangent of the slope's absolute value: \[\alpha = \tan^{-1}\left(\frac{5}{3}\right)\].This is the acute angle between the line and the negative \(x\)-axis. Since the line is in the second quadrant, \(\theta = 180^\circ - \alpha\). Calculate \(\alpha\) and \(\theta\) accordingly.

Step 4: Determine Trigonometric Functions

With \(\alpha\), say approximately \(59^\circ\), compute \(\theta = 180^\circ - 59^\circ = 121^\circ\). Now, determine the six trig functions of \(\theta\): 1. **Sine**: \(\sin(\theta) = \sin(59^\circ)\). 2. **Cosine**: \(\cos(\theta) = -\cos(59^\circ)\). 3. **Tangent**: \(\tan(\theta) = -\tan(59^\circ)\). 4. **Cosecant**: \(\csc(\theta) = \frac{1}{\sin(\theta)}\). 5. **Secant**: \(\sec(\theta) = -\frac{1}{\cos(\theta)}\). 6. **Cotangent**: \(\cot(\theta) = -\frac{1}{\tan(\theta)}\).

Step 5: Compute the Exact Values using Trigonometric Identities

Assuming the terminal side's coordinates as \((-3, 5)\), use \(r = \sqrt{x^2 + y^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}\).We can now evaluate:- \(\sin(\theta) = \frac{5}{\sqrt{34}}\)- \(\cos(\theta) = -\frac{3}{\sqrt{34}}\)- \(\tan(\theta) = -\frac{5}{3}\)- \(\csc(\theta) = \frac{\sqrt{34}}{5}\)- \(\sec(\theta) = -\frac{\sqrt{34}}{3}\)- \(\cot(\theta) = -\frac{3}{5}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using a coordinate system. It combines algebra and geometry to gain insights into the shapes and sizes of different geometric figures. In this problem, coordinate geometry helps us to find the equation of the line and understand the position of the terminal side of the angle in relation to the coordinate axes.

By examining the equation \(-5x - 3y = 0\), we transform it into the slope-intercept form: \(y = -\frac{5}{3}x\). This allows us to determine the slope of the line, which is crucial for understanding the tilt of the line. Since the equation specifies \(x \leq 0\), it tells us that the line is positioned in the left portion of the coordinate grid, specifically in the second quadrant.

Coordinate Geometry is essential for determining where lines intersect, how they are positioned, and how they interact with axes. This foundational knowledge is vital for visualizing and solving trigonometric problems.

Angle Measurement

Measuring angles in geometry is fundamental for understanding trigonometric relationships. Angles can be measured in degrees or radians. The problem at hand uses degrees to express the angle \(\theta\) in standard position.

Standard position is when the angle's vertex is at the origin of the coordinate system and its initial side lies on the positive x-axis. The angle then opens counter-clockwise to reach the terminal side. For this problem, the terminal side of the angle is on the line given by the equation \(y = -\frac{5}{3}x\).

The measurement of the angle \(\theta\), in this case, involves finding the angle between the line and the x-axis. By using the arctangent of the absolute value of the slope, we find the reference angle. Adjustments are made to account for its position in the second quadrant, where angles are measured from 180° backward towards the positive x-axis.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation so that the slope and y-intercept are immediately identifiable. This form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.

In this exercise, the original equation \(-5x - 3y = 0\) was converted into the slope-intercept form \(y = -\frac{5}{3}x\). The y-intercept is not explicitly visible in this case as the line passes through the origin (0,0).

The slope, \(-\frac{5}{3}\), dictates how steep the line is and the direction it takes. A negative slope indicates that as we move along the x-axis in a positive direction, the y-value decreases. This form of the equation allows us to predict and understand how changes in x affect the y values along the line. Recognizing the slope's value helps us to find the angle the line makes with the x-axis using the tangent function.

Reference Angles

Understanding reference angles is key to dealing with trigonometric functions, especially in non-standard quadrants. A reference angle is the acute angle that a line makes with the nearest x-axis. In trigonometry, this helps in determining the values of trigonometric functions based on known values in the first quadrant.

In the context of this problem, the reference angle \(\alpha\) is determined by the arctangent of the slope's absolute value: \(\alpha = \tan^{-1}\left(\frac{5}{3}\right)\). This is the angle between the line and the negative x-axis, measured within the right angle.

Since \(\theta\) is in the second quadrant, the angle is calculated as \((180^\circ - \alpha)\). The result gives us the actual angle needed to compute the six trigonometric functions. Reference angles simplify the computation of trigonometric functions, ensuring that correct signs and values are applied relative to the quadrant.