Find the exact values of the six trigonometric functions of θif the terminal side of θin standar

step 1 Define our 6 -truthmental function based with negative 1 -0. So first of all, we have our sine o

Find the exact values of the six trigonometric functions of...

Key Concepts

Standard Position and Terminal Side

In trigonometry, an angle is typically drawn in standard position, meaning its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side of an angle is the ray where the angle ends, and its location on the coordinate plane provides the x and y values that are used to define the trigonometric functions.

Coordinate Definitions of Trigonometric Functions

When a point (x, y) lies on the terminal side of an angle, the six trigonometric functions can be defined in terms of x, y, and r, where r is the distance from the origin to the point. Specifically, sine is defined as y/r, cosine as x/r, and tangent as y/x, with the other functions (cosecant, secant, and cotangent) being the respective reciprocals of sine, cosine, and tangent.

Using the Pythagorean Theorem to Find r

The value of r, which represents the distance from the origin to a given point (x, y) on the terminal side, is calculated using the Pythagorean theorem. The formula r = ?(x² + y²) is crucial because it normalizes the x and y coordinates, thereby allowing the correct evaluation of the trigonometric functions.

Reciprocal Trigonometric Functions

Based on the primary definitions of sine, cosine, and tangent, the reciprocal functions are defined as follows: cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. These definitions are important for comprehensively understanding trigonometry, as they extend the family of trigonometric functions.

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