Problem 43 - 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 \(\theta\). $$x+y=0, x \geq 0$$

Step-by-Step Solution

Answer

The least positive angle \(\theta\) is \(315^\circ\). The trigonometric values are: \(\cos(\theta) = \frac{1}{\sqrt{2}}\), \(\sin(\theta) = -\frac{1}{\sqrt{2}}\), \(\tan(\theta) = -1\), \(\sec(\theta) = \sqrt{2}\), \(\csc(\theta) = -\sqrt{2}\), \(\cot(\theta) = -1\).

Step 1: Determine the Equation of the Line

The given equation is \(x + y = 0\). This can be rewritten in slope-intercept form as \(y = -x\). This is a line that passes through the origin with a slope of \(-1\).

Step 2: Find the Intersection with Restrictions

The restriction \(x \geq 0\) limits the line to the region in the coordinate plane where \(x\) is non-negative. This corresponds to the line segment in the first and fourth quadrants, specifically along the line from the origin at an angle of \(-45^\circ\) or \(315^\circ\) when measured counterclockwise from the positive \(x\)-axis.

Step 3: Identify the Least Positive Angle

To determine the least positive angle \(\theta\), we measure it from the positive \(x\)-axis in a counterclockwise direction. The angle is \(315^\circ\) since this is the positive angle that reaches the terminal side of the angle described by the line \(y=-x\), given the restriction.

Step 4: Calculate the Trigonometric Functions

Since the terminal side is at an angle of \(315^\circ\) or \(-45^\circ\), we use the properties of a 45-degree reference angle. For an angle \(\theta=315^\circ\):\[ \cos(\theta) = \frac{1}{\sqrt{2}}, \quad \sin(\theta) = -\frac{1}{\sqrt{2}} \]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = -1 \]\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \sqrt{2}, \quad \csc(\theta) = \frac{1}{\sin(\theta)} = -\sqrt{2} \]\[ \cot(\theta) = \frac{1}{\tan(\theta)} = -1 \]

Step 5: Validate the Trigonometric Values

In the fourth quadrant, \(\cos(\theta)\) remains positive, \(\sin(\theta)\) is negative, and \(\tan(\theta)\) is negative, which matches the properties seen in the calculations.

Key Concepts

Angle Measurement

In trigonometry, understanding how to measure angles is crucial. Angles can be measured in degrees or radians, and their size dictates the position and direction of a vector in relation to a fixed point. The angle measurement starts from a reference line, typically the positive x-axis, and extends counterclockwise around the unit circle.
When measuring angles:

• Full rotation is 360 degrees or \(2\pi\) radians.
• Angles can be positive or negative.
• Positive angles result from counterclockwise rotation.
• Negative angles result from clockwise rotation.

For example, the angle of 315 degrees is equivalent to \(-45\) degrees, as it can be measured counterclockwise from the positive x-axis or clockwise passing the negative x-axis.

Coordinate Plane

The coordinate plane, or Cartesian plane, consists of two perpendicular lines, the x-axis and the y-axis, which intersect at a point called the origin. An angle is often measured from the positive x-axis along a circle centered on the origin to determine its position in the plane.

• Quadrants divide the plane into four sections.
• The first quadrant is both x and y positive.
• The second quadrant has x negative and y positive.
• The third quadrant is x and y both negative.
• The fourth quadrant has x positive and y negative.

Trigonometric functions vary depending on the quadrant. For instance, in the fourth quadrant, as in our exercise, cosine is positive while sine and tangent remain negative. This is critical when determining the proper sign of trigonometric functions based on the angle's location.

Reference Angle

A reference angle assists in simplifying expressions for trigonometric functions. It is the smallest angle from the terminal side of an angle to the x-axis, always measured as a positive angle.
In any quadrant:

• The reference angle is always less than or equal to 90 degrees.
• For angles in the fourth quadrant, the reference angle can be found by subtracting the angle from 360 degrees.
• In our exercise, the angle 315 degrees has a reference angle of \(45\) degrees, as \(360 - 315 = 45\).

This reference angle allows us to apply known trigonometric ratios of standard angles (like 45 degrees) to any angle in the given quadrant, making calculations straightforward.

Standard Position

The concept of standard position pertains to an angle positioned on the coordinate plane where its vertex is at the origin, and its initial side lies along the positive x-axis. Measuring angles from this initial position helps provide a uniform method of defining trigonometric functions.
Key points of the standard position:

• All angles are assumed to be measured from the positive x-axis.
• This consistent starting point helps in defining the direction and magnitude of the angle.
• Angles in standard position enable us to categorize them easily into the four quadrants based on their terminal side location.

For example, an angle of 315 degrees measured counterclockwise places its terminal side in the fourth quadrant, aligning with our equation \(x + y = 0\) and complementing our trigonometric calculations.