Problem 45 - 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 \(\theta\). $$-\sqrt{3} x+y=0, x \leq 0$$

Step-by-Step Solution

Answer

The angle θ is 120°, with trigonometric values: sin 120°=√3/2, cos 120°=-1/2, tan 120°=-√3, csc 120°=2√3/3, sec 120°=-2, cot 120°=-√3/3.

Step 1: Convert the given line equation into slope-intercept form

The given line equation is \(-\sqrt{3}x + y = 0\). We can rewrite it in the slope-intercept form \(y = mx + b\) by solving for \(y\):\[-\sqrt{3}x + y = 0\]Add \(\sqrt{3}x\) to both sides:\[y = \sqrt{3}x\]The equation of the line is \(y = \sqrt{3}x\). Here, the slope \(m = \sqrt{3}\) and the y-intercept \(b = 0\).

Step 2: Understand the restriction on x

The restriction \(x \leq 0\) indicates that the terminal side of angle \(\theta\) is on the left side of the y-axis in the Cartesian coordinate system.

Step 3: Determine the angle θ from the slope

The line \(y = \sqrt{3}x\) has a positive slope of \(\sqrt{3}\), which corresponds to a reference angle of \(\theta = 60^\circ\) (in quadrant I). However, given \(x \leq 0\), the terminal side lies in quadrant II. The angle in standard position is \(180^\circ - 60^\circ = 120^\circ\). Thus, \(\theta = 120^\circ\).

Step 4: Calculate the six trigonometric functions of θ

Use the angle \(\theta = 120^\circ\) to find the values:- Sine: \( \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \)- Cosine: \( \cos(120^\circ) = \cos(180^\circ - 60^\circ) = -\cos(60^\circ) = -\frac{1}{2} \)- Tangent: \( \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) = -\sqrt{3} \)- Cosecant: \( \csc(120^\circ) = \frac{1}{\sin(120^\circ)} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)- Secant: \( \sec(120^\circ) = \frac{1}{\cos(120^\circ)} = -2 \)- Cotangent: \( \cot(120^\circ) = \frac{1}{\tan(120^\circ)} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \)

Key Concepts

Angle Measurement

In trigonometry, angles are measured in degrees or radians. Understanding angle measurement is crucial for correctly interpreting problems and equations. Typically, angles are measured from the positive x-axis in a counter-clockwise direction. This is known as the 'standard position' of an angle. When working with angles, it's also essential to grasp the concept of reference angles. A reference angle is the acute angle that an angle makes with the x-axis. Thus, every angle in a coordinate plane has a reference angle that helps describe its position in terms of a right-angled triangle.

For instance, in the given exercise, we started with the slope of a line to find our reference angle as 60°. However, because of the restriction on x that places our angle in the second quadrant, the angle θ becomes 120°, since it takes a position on the negative side of the x-axis.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. They are foundational to solving trigonometry problems. The six primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Each of these functions is associated with a ratio of sides in a right triangle.

• Sine (\(\sin\theta\)): opposite side over hypotenuse
• Cosine (\(\cos\theta\)): adjacent side over hypotenuse
• Tangent (\(\tan\theta\)): opposite side over adjacent side
• Cosecant (\(\csc\theta\)): reciprocal of sine
• Secant (\(\sec\theta\)): reciprocal of cosine
• Cotangent (\(\cot\theta\)): reciprocal of tangent

In our example, at 120°, these functions compute to:

• \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\)
• \(\cos(120^\circ) = -\frac{1}{2}\)
• \(\tan(120^\circ) = -\sqrt{3}\)
• \(\csc(120^\circ) = \frac{2\sqrt{3}}{3}\)
• \(\sec(120^\circ) = -2\)
• \(\cot(120^\circ) = -\frac{\sqrt{3}}{3}\)

Understanding these functions allows us to solve a wide range of problems in trigonometry.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane where every point is determined by a pair of numerical coordinates. These coordinates are the distances from two fixed perpendicular lines, called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. This system allows us to represent geometric figures and perform algebraic operations graphically.

Knowing how to interpret positions and movements within this system is essential, especially in trigonometry. Quadrants are the four sections of the Cartesian plane, each determined by the signs of the x and y coordinates. By understanding these quadrants:

• Quadrant I has positive x and y values
• Quadrant II has negative x and positive y values
• Quadrant III has negative x and y values
• Quadrant IV has positive x and negative y values

In our exercise, the restriction of \(x \leq 0\) informs us that the angle’s terminal side lies in Quadrant II. Understanding this helps correctly determine angle characteristics and the sign of trigonometric functions.