Trigonometric Functions of Real Numbers

The unit circle, centered at the origin, defines the trigonometric functions.

Each real number represents an arc length measured from the positive x-axis and rotate it counterclockwise around the circle.

The x-coordinate gives the cosine of the real number. Similarly, the y-coordinate gives the sine.

Tangent is derived as sine over cosine, cotangent is cosine over sine, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.

Dividing the circle into four quadrants helps identify a function’s sign.

All functions are positive in the first quadrant, where x and y are positive. In the second, since x is negative, the cosine is negative. As y is positive, sine is also positive. Tangent, which is y over x, is negative because a positive over a negative is negative.

In the third, tangent and cotangent are positive, as x and y are negative. In the fourth, only cosine and secant are positive, as x is positive and y is negative.

Trigonometric functions are useful for real-world calculations, such as estimating a building’s height by multiplying the known horizontal distance by the tangent of the known angle of inclination.