Problem 44 - 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 \(\pi/4\), with trig functions: \(\sin=\frac{\sqrt{2}}{2}\), \(\cos=\frac{\sqrt{2}}{2}\), \(\tan=1\), \(\csc=\sqrt{2}\), \(\sec=\sqrt{2}\), \(\cot=1\).

Step 1: Identify the Terminal Side of the Angle

The equation given is \(x - y = 0\), which can also be written as \(y = x\). This represents a line through the origin with a slope of 1, which is a line at a 45-degree angle from the positive x-axis.

Step 2: Determine the Least Positive Angle \(\theta\)

The line \(y = x\) forms an angle of 45 degrees with the positive x-axis in the coordinate plane. In radians, this angle is equal to \(\pi/4\). There are no negative restrictions on \(x\), so the least positive angle \(\theta\) is \(\pi/4\).

Step 3: Calculate the Trigonometric Functions

For \(\theta = \pi/4\), we calculate the trigonometric functions using the special values of a 45° angle: * \(\sin \theta = \sin(\pi/4) = \frac{\sqrt{2}}{2}\) * \(\cos \theta = \cos(\pi/4) = \frac{\sqrt{2}}{2}\) * \(\tan \theta = \tan(\pi/4) = 1\) * \(\csc \theta = \csc(\pi/4) = \sqrt{2}\) * \(\sec \theta = \sec(\pi/4) = \sqrt{2}\) * \(\cot \theta = \cot(\pi/4) = 1\)

Step 4: Verify the Values

Double-check each calculated trigonometric value for \(\theta = \pi/4\) and confirm their correctness using known trigonometric identities, such as \(\sin^2 \theta + \cos^2 \theta = 1\). This identity holds as \(\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = 1\).

Key Concepts

Terminal Side of an Angle

Understanding the terminal side of an angle is crucial in trigonometry. When an angle is drawn on a coordinate plane, it starts at the initial side, often along the positive x-axis, and rotates towards its terminal side. This terminal side can lie in any of the four quadrants, depending on the angle's measure. In the equation given, \( x - y = 0 \), which is equivalent to \( y = x \), the terminal side lies along this line in the plane.
Because this line passes through the origin and has a slope of 1, it makes a 45-degree angle with the positive x-axis. In this context, the terminal side is a line extending from the origin outward through the first quadrant. This is the precise location where the angle reaches its degrees of rotation. Remember, the significance of understanding where the terminal side lies helps in identifying the angle correctly and determining the relevant trigonometric function values.

Standard Position Angle

A standard position angle is an angle whose vertex is at the origin and whose initial side lies along the positive x-axis. This concept simplifies understanding angles in the coordinate plane because it provides a consistent starting point. For the exercise, angle \( \theta \) starts on the positive x-axis and rotates counterclockwise to align with the line \( y = x \).
This rotation results in an angle of 45 degrees, or \( \pi/4 \) radians, which is one-quarter of a right angle. It's considered a positive angle because the direction is counterclockwise. Knowing an angle in standard position helps when solving for its trigonometric functions, as you can then apply special values and identities to predict outcomes effectively.

Special Angle Values

Special angle values refer to angles with known trigonometric function values, typically 30°, 45°, and 60° (or equivalents in radians). For these angles, the sine, cosine, and tangent values, along with their reciprocals, have specific, easily memorizable values.
For the 45-degree angle in this exercise, these values are repeated due to symmetry. At \( \theta = \pi/4 \):

• \( \sin \theta = \frac{\sqrt{2}}{2} \)
• \( \cos \theta = \frac{\sqrt{2}}{2} \)
• \( \tan \theta = 1 \)
• \( \csc \theta = \sqrt{2} \)
• \( \sec \theta = \sqrt{2} \)
• \( \cot \theta = 1 \)

These values originate from the properties of an isosceles right triangle, where the two non-hypotenuse sides are equal. These angles form the foundation of many trigonometric computations due to their simplicity and regularity, making them often used as reference angles for deeper analysis.