The point is on the terminal side of an angle in standard position. Determine the exact values o

step 1 We have r is equal to square root of x square plus y square, which is equal to square root of 5

The point is on the terminal side of an angle in standard...

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, and their reciprocals, are defined using the coordinates of a point on the terminal side and the distance r. Sine is the ratio of the y?coordinate to r, cosine is the ratio of the x?coordinate to r, and tangent is the ratio of the y?coordinate to the x?coordinate. These definitions extend to secant, cosecant, and cotangent as the reciprocals of cosine, sine, and tangent respectively.

Quadrant Analysis

The coordinate plane is divided into four quadrants, and the sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies. Understanding which quadrant the angle terminates in is essential, as it determines the positive or negative values of the x?coordinate, y?coordinate, and therefore of the trigonometric ratios.

Pythagorean Theorem

The Pythagorean theorem is used to calculate the distance (r) from the origin to the point on the terminal side of the angle by forming a right triangle with the x?axis. This distance, given by r = ?(x² + y²), is crucial for defining the sine, cosine, and other trigonometric functions in terms of the coordinates.

Terminal Side and Coordinates

The terminal side of an angle in standard position is the ray that extends from the origin and forms the angle with the positive x?axis. When a specific point lies on this terminal side, its coordinates (x, y) are used to form a right triangle with the origin, enabling the determination of the basic trigonometric ratios.

Standard Position of an Angle

In trigonometry, an angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x?axis. This concept is fundamental because it allows for a consistent way to define and compute the trigonometric functions based on the point where the terminal side intersects the coordinate plane.

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