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. (―8 , 15)

Verified step by step guidance

1. insert step 1: Plot the point (-8, 15) on the Cartesian coordinate system. This point is on the terminal side of the angle θ in standard position, where the initial side is along the positive x-axis.
2. insert step 2: Determine the reference angle by using the tangent function. The reference angle can be found using \( \tan^{-1}\left(\frac{15}{-8}\right) \).
3. insert step 3: Since the point (-8, 15) is in the second quadrant, the angle θ is \( 180^\circ - \text{reference angle} \).
4. insert step 4: Calculate the hypotenuse (r) using the Pythagorean theorem: \( r = \sqrt{(-8)^2 + 15^2} \).
5. insert step 5: Use the coordinates and the hypotenuse to find the six trigonometric functions: \( \sin(\theta) = \frac{15}{r} \), \( \cos(\theta) = \frac{-8}{r} \), \( \tan(\theta) = \frac{15}{-8} \), \( \csc(\theta) = \frac{r}{15} \), \( \sec(\theta) = \frac{r}{-8} \), \( \cot(\theta) = \frac{-8}{15} \).

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 angle is measured counterclockwise from the initial side. Understanding this concept is crucial for accurately sketching the angle θ and determining its terminal side based on the given point.

Coordinates and the Terminal Side

The terminal side of an angle in standard position can be determined by the coordinates of a point on that side. For the point (−8, 15), the angle θ is formed by the line connecting the origin to this point. This relationship helps in visualizing the angle and calculating the trigonometric functions based on the coordinates.

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined based on the ratios of the sides of a right triangle or the coordinates of a point on the unit circle. For the point (−8, 15), these functions can be calculated using the formulas: sin(θ) = y/r, cos(θ) = x/r, and tan(θ) = y/x, where r is the distance from the origin to the point, providing a complete set of values for the angle.