Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

Verified step by step guidance

1. Step 1: Plot the point (-2\sqrt{3}, 2) on the Cartesian coordinate system. This point lies in the second quadrant because the x-coordinate is negative and the y-coordinate is positive.
2. Step 2: Draw a line from the origin (0,0) to the point (-2\sqrt{3}, 2). This line represents the terminal side of the angle \(\theta\) in standard position.
3. Step 3: Calculate the radius (r) using the distance formula: \(r = \sqrt{(-2\sqrt{3})^2 + 2^2}\). This will be used to find the trigonometric functions.
4. Step 4: Determine the six trigonometric functions: \(\sin(\theta) = \frac{y}{r}\), \(\cos(\theta) = \frac{x}{r}\), \(\tan(\theta) = \frac{y}{x}\), \(\csc(\theta) = \frac{r}{y}\), \(\sec(\theta) = \frac{r}{x}\), \(\cot(\theta) = \frac{x}{y}\).
5. Step 5: Simplify each trigonometric function and rationalize the denominators where applicable.

Key Concepts

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The terminal side of the angle is determined by rotating the initial side counterclockwise for positive angles and clockwise for negative angles. Understanding this concept is crucial for accurately sketching the angle θ and identifying its terminal side based on the given point.

Coordinates and Trigonometric Functions

The coordinates of a point on the terminal side of an angle in standard position can be used to find the values of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. For a point (x, y), these functions are defined as sin(θ) = y/r, cos(θ) = x/r, and tan(θ) = y/x, where r is the distance from the origin to the point, calculated as r = √(x² + y²). This relationship is essential for solving the problem.

Rationalizing Denominators

Rationalizing the denominator involves rewriting a fraction so that there are no radical expressions in the denominator. This is often done by multiplying the numerator and denominator by a suitable value that eliminates the radical. In trigonometry, this is important for presenting the values of trigonometric functions in a standard form, making them easier to interpret and use in further calculations.