a point on the terminal side of an angle θin standard position is given. Find the exact value of

step 1 A point on the terminal side of an angle theta in a position is given, which is negative 3, nega

A point on the terminal side of an angle θin standard...

Key Concepts

Standard Position

An angle is said to be in standard position when its vertex is at the origin of the Cartesian coordinate system and its initial side lies along the positive x-axis. This positioning is crucial as it provides a consistent framework for defining trigonometric functions based on the endpoint of the angle's terminal side.

Terminal Side and Coordinate Points

The point on the terminal side of the angle represents the coordinates (x, y) in the Cartesian plane. These coordinates are used to determine the signs and magnitudes of the trigonometric functions. Knowing the location of the point relative to the coordinate axes is essential for properly evaluating the functions.

Distance from the Origin (r) Calculation

The distance from the origin to the point on the terminal side, denoted by r, is calculated using the Pythagorean theorem as r = ?(x² + y²). This distance acts as the hypotenuse in the right triangle formed by dropping a perpendicular from the point to the x-axis and is central to defining the sine and cosine functions in a geometric context.

Pythagorean Theorem

The Pythagorean theorem is used to relate the coordinates of the point (x, y) to the hypotenuse (r) in a right triangle through the relationship x² + y² = r². This theorem is the foundation for finding the exact value of r and, by extension, the exact values of the trigonometric functions.

Trigonometric Function Definitions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined in terms of the coordinates (x, y) and the distance r. Specifically, sine is y/r, cosine is x/r, and tangent is y/x, with the remaining functions being their respective reciprocals. These definitions allow one to compute the exact value of each function for any angle in standard position.

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