For the given angle θcorresponding to the point P(-4,-3) in the figure, evaluate sinθ, cosθ, tan

For the given angle θcorresponding to the point P(-4,-3) in...

Key Concepts

Cartesian Coordinate Representation

In trigonometry, points in the Cartesian plane are used to represent angles and their associated trigonometric values. A point (x, y) on the terminal side of an angle in standard position helps determine the sine, cosine, and tangent through its coordinates, linking algebraic positions to geometric angles.

Distance Formula and the Radius (r)

The distance from the origin to the point (x, y) is calculated using the formula r = ?(x² + y²). This value is crucial because it represents the hypotenuse of the right triangle formed by dropping a perpendicular from the point to the x-axis, and it is used to define the trigonometric functions as ratios involving x, y, and r.

Definition of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are defined using the coordinates (x, y) on the unit circle or any circle of radius r. Specifically, sin ? = y/r, cos ? = x/r, and tan ? = y/x, with additional reciprocal functions defined as sec ? = 1/cos ?, csc ? = 1/sin ?, and cot ? = 1/tan ?.

Quadrant Analysis and Sign Determination

Identifying the quadrant in which the point lies is critical for determining the correct sign (positive or negative) of trigonometric functions. Since the coordinates x and y determine the position relative to the axes, each quadrant has a specific pattern of signs for sine, cosine, and tangent, which must be observed when evaluating these functions.