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Secondary Circular Functions
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The three primary circular functions, as already defined, form the cornerstone of trigonometry Three more circular functions exist Their values represent the reciprocals of the values of the primary circular functions
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The cosecant function Imagine the ray OP in Fig 2-1, oriented at a certain angle q with respect to the x axis, pointing outward from the origin, and intersecting the unit circle at P = (x0,y0) The reciprocal of the ordinate, 1/y0, is defined as the cosecant of the angle q The cosecant function is abbreviated as csc, so we can write
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csc q = 1/y0 Because y0 is the value of the sine function, the cosecant is the reciprocal of the sine For any angle q, the following equation is always true as long as sin q 0: csc q = 1/sin q The cosecant of an angle q is undefined when q is any integer multiple of p That s because the sine of any such angle is 0, which would make the cosecant equal to 1/0 Figure 2-6 is a graph of the cosecant function for values of q between 3p and 3p The vertical dashed lines denote the singularities There s also a singularity along the y axis
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The secant function Consider the reciprocal of the abscissa, that is, 1/x0, in Fig 2-1 This value is the secant of the angle q The secant function is abbreviated as sec, so we can write
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sec q = 1/x0 The secant of an angle is the reciprocal of the cosine When cos q 0, the following equation is true: sec q = 1/cos q The secant is undefined for any positive or negative odd-integer multiple of p /2 Figure 2-7 is a graph of the secant function for values of q between 3p and 3p Note the input values for which the function is singular (vertical dashed lines)
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Secondary Circular Functions
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Figure 2-6 Graph of the cosecant function for values of q between 3p and 3p
Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
sec q
3p 1
q 3p
Figure 2-7 Graph of the secant function for values of q between 3p and 3p
Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
A Fresh Look at Trigonometry
The cotangent function Now let s think about the value of x0 /y0 at the point P where the ray OP crosses the unit circle This ratio is called the cotangent of the angle q The cotangent function is abbreviated as cot, so we can write
cot q = x0 /y0 Because we already know that cos q = x0 and sin q = y0, we can express the cotangent function in terms of the cosine and the sine: cot q = cos q/sin q The cotangent function is also the reciprocal of the tangent function: cot q = 1/tan q Whenever y0 = 0, the denominators of all three quotients above become 0, so the cotangent function is not defined Singularities occur at all integer multiples of p Figure 2-8 is a graph of the cotangent function for values of q between 3p and 3p Singularities are, as in the other examples here, shown as vertical dashed lines
cot q 3 2 1
3p 1 2 3
q 3p
Figure 2-8 Graph of the cotangent function for values of q between 3p and 3p Each division
on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
Pythagorean Extras
Are you confused
Now that you know how the six circular functions are defined, you might wonder how you can determine the output values for specific inputs The easiest way is to use a calculator This approach will usually give you an approximation, not an exact value, because the output values of trigonometric functions are almost always irrational numbers Remember to set the calculator to work for inputs in radians, not in degrees! The values of the sine and cosine functions never get smaller than 1 or larger than 1 The values of the other four functions can vary wildly Put a few numbers into your calculator and see what happens when you apply the circular functions to them When you input a value for which a function is singular, you ll get an error message on the calculator
Here s a challenge!
Figure out the value of cot (5p /4) As in the previous challenge, you should be able to solve this problem entirely with geometry
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