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Topic: **What are the problem solving methods****Question:**
I would like to know how to determine the method used to solve certain trig problems.. My instructor doesn't give me much information on the subject. Example:
Use periodic function to find the value of cot 15pie/2.
Evaluate the quadrantal angle csc pie.
I need to know how to tell what type operation needs to be used to find the different Trig functions of an angle.. Any help?

July 16, 2019 / By Claribel

Your instructor appears to be trying to make the point that periodic functions are, periodic. So the cot(15PI/2) = cot(6PI + 3PI/2) = cot(3PI/2). Since each 2PI we add simply increments the independent variable by one full period, we can divide out and drop the even multiples, using only the remainder to find the value of the function. And, since cot = cos/sin we may also use what we know of the values for cos and sin at 3PI/2 to find the answer. sin(3PI/2) = -1, and cos(3PI/2) = 0 so cot(3PI/2) = 0 = cot(15PI/2) = cot(3PI/2 + 2nPI) where n is any integer. csc(PI) = 1/sin(PI) which is undefined, since sin(PI) = 0. csc(PI + 2nPI) for any integer value n will be undefined. You can translate between real values (radians) and angles as arguments for the trigonometric functions using 2PI = 360 degrees. So PI = 180 degrees and 3PI/2 = 270 degrees, but this can be confusing if you are thinking of the trig functions as being defined as ratios of the sides of right triangles, since you cannot draw triangles with angles of 180 or 270 degrees. To appreciate them as periodic functions you must really use a definition that yeilds values for all real numbers (except in those cases for tan, cot, sec and csc where they are undefined). Generally this is done using a unit circle centered at the origin, where the independent variable in radians is the distance from the point on the circle at (1,0), measured along the arc (including any number of full circles equal to 2PI). IF we let w be this independent variable, and the point on the circle w radians from (1,0) be represented by (xsubw, ysubw), then sin(w) = ysubw and cos(w) = xsubw tan(w) = ysubw/xsubw, cot = xsubw/ysubw etc. Of course, when the denominator is zero, the function is not defined. This definition makes the functions easy to determine for quadrantal angles like 90 degrees (0 , 1) or 180 degrees (-1, 0), or 270 degrees (0, -1).

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First the definitions of Sine, Cosine and Tangent. Consider a right angled triange ABC where B is the right angle and has sides of a, b and c opposite the angles at A, B and C This means that the hypotenuse is 'b' SineA = [opposite side]/[hypotenuse] = a/b CosineA = [adjacent side]/[hypotenuse] = c/b Now consider a right angle triangle with each of the other angles equal to 45 degrees. Call the two equal sides of length '1'. Then by Pythagoras' Theorem the hypotenuse is equal to sqrt(2) and Sin45 = 1/[sqrt(2)] Cos45 = 1/[sqrt(2)] also Now consider an equalateral triangle each side equal to two units and each of its angles equal to 60 degrees. Drop a perpendicular from the uppermost angle on to the base. This has the effect of generating what is known as a 60, 30, 90 triangle with the side opposite the right angle that is to say the hypotenuse being equal to '2', the side opposite the 30 degee angle equat to '1' and the side opposite the 60 degree angle equal to sqrt(3) This means that Sin60 = [sqrt(3)]/2 Cos60 = 1/2 Sin30 = 1/2 Cos30 = [sqrt(3)]/2 Therefore Sin45 + Cos60 = 1/[sqrt(2)] + 1/2 = [sqrt(2)]/2 +1/2 = {1 + [sqrt(2)]}/2 Consider 'The Unit Circle' and use the earlier definitions to determine sin and cos of angles greater than 90 degrees - you must allow for negative values. Cos180 degrees is equal to -1 and Sin180 degrees is equal to zero Therefore Cos180 - Sin180 = -1 - 0 = -1

csc(x) is defined as being (1/sin(x), cot(x) is defined as (cos(x0 / sin(x)) csc(-3pi/4) = csc(5pi/4) = (1 / 5pi/4) = (-1 / sqrt(2)) = -sqrt(2) / 2. cot(pi/4) = (cos(pi/4) / sin(pi/4)) = (1/2) / (1/2) Answer 1 - sqrt(2)/2 If you want to check this on a calculator: (1) pi/4 = 45 degrees (2) -3pi/4 = -135 degrees and since you don't want to work with negative angles just add 360 degrees to and get 225 degrees. -135 degrees = 225 degrees and take it from there. .

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