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Nine Trigonometric FunctionsEmma BurginFST 10CMrs. DeweyDecember 19, 2011Throughout the past few chapters, much was learned about trigonometric functions. In chapter four, the class was reunited with a somewhat familiar topic from math classes past: the original trig functions, sine, cosine, and tangent. Chapter five then brought out new concepts that built off of the original functions: inverse trig functions and reciprocal trig functions. By the end of these chapters, a total of nine trig functions, all with their own specific characteristics and properties, had been discovered. Along with another familiar concept from previous courses, right triangle trigonometry, this essay will be about those nine trigonometric functions and the relationships between them. Figure 1. The Original Sine FunctionLet’s begin by talking about the first of the original trig functions: sine. This function is shown above in Figure 1. The function is a continuous wave with a domain such that x-values are any real number, positive and negative, stretching on forever. The function ranges between values of -1 and 1 on the y-axis. Because the function does not go past these values, these values are the maximum and minimum. Refer again to Figure 1 where there is a part of the function highlighted. This is one cycle of the sine function. This function is periodic. The cycle shown above ends where 2π lies along the x-axis. This means that the period of the function is 2π, because each cycle will end at a multiple of 2π. The final thing relating to the sine function is amplitude. This relates back to the maximum and minimum values. The maximum and minimum are 1 and -1, so that makes the amplitude of the original sine function 1. That’s all there is to it. Figure 2. The Original Cosine FunctionNext up on the list of original trig functions is cosine. This is an even function, meaning it has symmetry in respect to the y-axis and it is shown in Figure 2 above. This function is also a continuous wave with a domain of all real numbers and y-values ranging between -1 and 1. Like sine, the maximum and minimum values are 1 and -1 respectively, as these are the most extreme values the function reaches. Refer back to Figure 2 to see one cycle of the cosine function highlighted. This cycle of this periodic function shows that the period is also 2π, much like sine. Another likeness is this function’s amplitude of one. Cosine also has its own characteristics that make it unique. Figure 3. The Original Tangent FunctionThe third original trig function is tangent, an odd function shown in Figure 3. This function is a bit different than sine or cosine, who share many of the same characteristics. The range of this function is easy to see: the y-values will take on any real number. The domain is where things get a bit tricky. The x-values will take on most values with a few major exceptions. The graph of the original tangent function has vertical asymptotes anywhere along the x-axis where the value is an odd multiple of π/2. This means the domain of the tangent function is all real numbers except for odd multiples of π/2. Looking back up to Figure 3, a cycle of this function is highlighted. The graph shows how the cycle goes from –π/2 to π/2 (but never touching the asymptotes!) making the period of the tangent function π. In sine and cosine functions, what usually goes hand in hand with period is amplitude. However, tangent has no amplitude. This is because the range goes on forever and there is no maximum or minimum height to create the amplitude. The tangent function has properties not quite like sine and cosine, but that is what makes it one of the more unique trig functions. Figure 4. The Inverse Sine FunctionBranching off of the original trig functions are the inverse trig functions. The odd function of inverse sine, as represented by Figure 4, is shown above. This function must be restricted to be a function. If it was not restricted and the function was to be a complete wave, the graph would not be a function. (Recall how a graph must pass the “vertical line test,” showing that each input has only one output to be considered a function.) In the inverse sine function, values in the domain go from -1 to 1, and the function ranges from –π/2 to π/2. Sound familiar? It should, though not exactly. The domain and range of the original sine function, {x: x ∈ ?} and {y:-1≤y≤1}, are flip-flopped to give the domain and range of the inverse sine function (although remember about the restrictions and why the range can’t be all the real numbers.) A better way of referring to this transformation is a reflection. Simply put, the inverse sine function is a reflection of the original sine function over the y=x line. Figure 5. The Inverse Cosine FunctionThe next inverse trig function is inverse cosine. This function is shown above in Figure 5. Like inverse sine, the inverse cosine function is a reflection of the original cosine function over the y=x line. This function must also be restricted in order to pass the vertical line test and remain a function. Because this function is restricted, y-values only range between 0 and π, and values stay between -1 and 1 on the y-axis. This means that any value in this function belongs in the first or second quadrant. Also like the sine and inverse sine functions, the domain and range of the cosine function is flipped around and become the domain and range of the inverse cosine function. To summarize, the inverse cosine function is a reflection of cosine over the y=x line, it must be restricted to be a function, and the domain and range of the inverse cosine function is the domain and range of the cosine function switched around. Figure 6. The Inverse Tangent FunctionThe final inverse trig function is inverse tangent. This restricted odd function is shown in Figure 6. A look back at the original tangent function shows that this is a function with asymptotes. The same goes for the inverse tangent function, although it has horizontal asymptotes instead of vertical ones. These asymptotes lie where y= -π/2 and π/2. Because of these asymptotes, y-values cannot be –π/2 and π/2, the range of this function is {y: –π/2<y< π/2}. However, this function will span across the x-axis forever, so the domain of this function is {x: x ∈ ?}. Like the previous functions, the domain and range of inverse tangent are the flip-flopped domain and range of the original tangent function, because inverse tangent is a reflection of tangent over the y=x line. Figure 7. The Cosecant FunctionThe last variation of the original trig functions is the reciprocal trig functions. Once again, let’s begin with sine. The reciprocal of the sine function, also known as the cosecant function, is shown above in Figure 7. This is an odd function that, unlike the inverse trig functions, does not have to be restricted in order to be a function. This is not a continuous function because of the breaks in it and it does have vertical asymptotes where θ=0, π, 2π, and so on. The domain of this function is all real numbers, except for multiples of π (the asymptotes.) The range is all values greater than 1 and less than -1, and this is clearly visible by looking at the graph. The relationship this function has to the original sine function is that it is sine’s reciprocal, 1/sinθ.Figure 8. The Secant FunctionNext up is the reciprocal of the cosine function, shown in Figure 8. This function, also known as secant, has symmetry in respect to the y-axis, making it an even function. Like the cosecant function, this function as vertical asymptotes, but not in the same places. This function has asymptotes at odd multiples of π/2. This function has the same range as cosecant, y≥1 and y≤-1. The two functions have similar domains, but due to secant’s asymptotes being at odd multiples of π/2, the domain is all real values except for odd multiples of π/2. The relationship that this function has to the original cosine function is that it is the reciprocal of cosine, or 1/cosθ.Figure 9. The Cotangent FunctionThe last reciprocal trig function is, of course, cotangent, an odd function shown above in Figure 9. Cotangent is the reciprocal of the original tangent function. Unlike the original tangent function, this one has vertical asymptotes at multiples of π instead of odd multiples of π/2. The range of this function includes all real numbers, as it will continue to go up and down forever. The domain of this function includes all real numbers except for multiples of π, because the graph will never touch the asymptotes. The relationship the cotangent function has to the tangent function is that it is tangent’s reciprocal, or 1/tanθ. Figure 10. A Right TriangleAlthough there are no other types of functions that relate to the original trig functions, there is another concept that does: right triangle trigonometry. A right triangle with sides labeled h (for hypotenuse), o (for opposite of θ), and a (for adjacent to θ), and an angle labeled θ, is pictured above in Figure 10. This triangle comes with a set of trigonometric rations that help to determine sine, cosine, and tangent of θ. The ratio to determine sine of θ is sinθ=opposite/hypotenuse, opposite being the leg opposite of θ. If the opposite leg is 3 units and the hypotenuse is 5 units, sinθ would be equal to 3/5. The ratio that goes with cosine is adjacent/hypotenuse, adjacent being the leg adjacent to θ. Say the adjacent leg is 4 units and the hypotenuse is 5 units. Cosθ, based on the right triangle trig ratio, is equal to 4/5. The final trig ratio is for tangent. The ratio for this is opposite/adjacent. Using the examples of units previously mentioned, the opposite leg is 3 units and the adjacent leg is 4 units, so tanθ=3/4.Right triangle trig ratios don’t only apply to the original trig functions; they apply to reciprocal trig functions as well. The only difference is that the ratios of the reciprocal functions are the reciprocals of the original ratios. For example, where sinθ is opposite/hypotenuse, cscθ is hypotenuse/opposite. Referring back to the example units, cscθ=5/3. The same goes for secant. Where cosθ=adjacent/hypotenuse, secθ=hypotenuse/adjacent. For example, cosθ=4/5, so secθ=5/4. The same rule also applies to tangent. The ratio for tangent is opposite/adjacent. For example, tanθ=3/4, so cotθ=4/3. The ratios for the reciprocal functions are just the reciprocals of the original trig ratios. It really is as simple as that. In conclusion, much has been learned about trig functions and ratios. Not only are there plain old sine, cosine, and tangent functions, but there are inverse and reciprocal functions as well, and all have their own distinct properties and characteristics. The original and reciprocal functions can be tied in with right triangle trigonometry and everything relates back to those tree original functions. With familiar concepts and concepts some have only recently been introduced to, it’s safe to say that there is a lot to know about trigonometry and these functions. ................
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