The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) include the domain, range, period, asymptotes and amplitudes. The domain of a cosine and sine function is all real numbers and the range is -1 to 1. The period is 2π, and the amplitude is 1. They have no asymptotes. The domain of tangent is all real numbers except for π2+kπ. The range is all real numbers and the period is π. Tan has no amplitude and has asymptotes when x= π2+kπ.
The domain of a secant function is all real numbers except for π2+kπ. The domain of a cosecant function is all real numbers except for kπ. The range of both is (-∞.-1]U[1,∞) and the period is 2π. Secant has asymptotes when x=π2+kπ. Cosecant has asymptotes when x=kπ. They have no amplitude. Cotangent’s domain is all real numbers except for kπ. The range is all real numbers and the period is π. It has no amplitude and has asymptotes when x=kπ.
In an inverse function, the x coordinate, or the domain, and the y coordinate, the range, switch places. Since only one to one functions have inverses, we take the interval -π2 to π2, which contains all the possible values of the sine function. Now, the new domain is [-π2, π2], while the range stays the same. We then switch the domain and the range, so the domain and range of arcsin (x) is [-1,1] and [-π2, π2]. For cosine, the interval [0,π] contains all possible values, and the range is still [-1,1]. To find arcos (x) we invert the domain and range again, to get [-1,1] as the domain and [0,π] as the range. For arctan (x), the interval (-π2, π2) includes all possible values. The range still remains all real numbers. Exchanging the domain and range gives us all real numbers as the domain and (-π2, π2) as the range.
As you can see, the properties of the six trig functions have many similarities and the inverse trig functions’ domain and range can be obtained with the one to one property of inverse functions...
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