How Do You Find the Ratio of Sec Pi?


Quick Answer

If calculating the secant when using pi in radians, the ratio known as secant of pi, or sec pi, is equal to -1. The secant of pi is equivalent to 1/(cos pi).

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Full Answer

In right angle trigonometry, the secant of x is equal to the hypotenuse divided by the adjacent side; this ratio is the reciprocal of the cosine. For this reason, the secant of an angle theta can also be calculated as 1/(cos theta).

In the radian system, suppose one has a circle of radius one, centered at the origin (0,0). If an angle theta is made between the positive x axis, the origin and another line that intersects the unit circle at point (x,y), the cosine of theta is equal to the coordinate x, and the sine of theta is equal to the coordinate y.

A complete 360-degree revolution around a circle is equivalent to 2*pi radians in the radian system. Therefore, pi radians is equivalent to a half revolution around a circle, or 180 degrees. This angle of pi radians intersects the unit circle at the point (-1,0). Since the cosine of pi radians, or cosine of 180 degrees, is equal to the x coordinate of the intersection with the unit circle, the cosine of pi radians is equal to -1. Therefore, the secant of pi radians is 1/(-1) equals -1.

If pi were measured in degrees rather than radians, the secant of pi would be 1/cos(pi in degrees) equals 1/0.99849714986 equals 1.00150511209. However, it is not common to measure degrees in terms of pi.

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