An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . 12x + 5y = 0 , x ≥ 0 .

Verified step by step guidance

1. Rewrite the given equation of the terminal side in slope-intercept form to understand the line better. Starting with \(12x + 5y = 0\), solve for \(y\) to get \(y = -\frac{12}{5}x\).
2. Recognize that the terminal side of angle \(\theta\) lies along the line \(y = -\frac{12}{5}x\) with the restriction \(x \geq 0\). This means the terminal side is in the fourth quadrant or on the positive x-axis.
3. Find the reference angle \(\alpha\) by considering the slope as the tangent of the angle the line makes with the positive x-axis. Use \(\tan(\alpha) = \left| -\frac{12}{5} \right| = \frac{12}{5}\), then find \(\alpha = \arctan\left(\frac{12}{5}\right)\).
4. Determine the least positive angle \(\theta\) in standard position. Since the line is in the fourth quadrant (because \(x \geq 0\) and \(y\) is negative), calculate \(\theta = 2\pi - \alpha\) (in radians) or \(360^\circ - \alpha\) (in degrees).
5. Calculate the six trigonometric functions of \(\theta\) by first choosing a point on the terminal side that satisfies the line equation and \(x \geq 0\). For example, use \(x = 5\), then \(y = -12\). Compute the hypotenuse \(r = \sqrt{x^2 + y^2} = \sqrt{5^2 + (-12)^2}\). Then use the definitions: \(\sin \theta = \frac{y}{r}\), \(\cos \theta = \frac{x}{r}\), \(\tan \theta = \frac{y}{x}\), \(\csc \theta = \frac{r}{y}\), \(\sec \theta = \frac{r}{x}\), and \(\cot \theta = \frac{x}{y}\).

Key Concepts

Standard Position of an Angle

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side is determined by rotating the initial side by the angle θ. Understanding this helps in visualizing and sketching the angle based on the given line equation.

Equation of a Line and Its Relation to an Angle

The equation 12x + 5y = 0 represents a line through the origin. The slope of this line corresponds to the tangent of the angle θ formed by the terminal side with the positive x-axis. The restriction x ≥ 0 limits the terminal side to the right half-plane, ensuring the angle is the least positive angle.

Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of the sides of a right triangle or coordinates on the unit circle. Once θ is identified, these functions can be calculated using the coordinates of a point on the terminal side or the slope of the line.