Lesson REFERENCE ANGLES AND HOW THEY'RE USED
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angles can be positive or negative. they can extend from 0 to infinity and from 0 to - infinity. it is general practice, however, to determine your reference angle from equivalent angles that are greater than or equal to 0 degrees and less than 360 degrees. you can convert any angle to an equivalent angle by doing the following. IF THE ORIGINAL ANGLE IS GREATER THAN OR EQUAL TO 360 DEGREES if you are given an angle that is greater than or equal to 360 degrees then you divide the angle by 360 degrees and then multiply 360 degrees by the integer part of the answer and then subtract the result of that from the original angle to get the equivalent angle that will be greater than or equal to 0 degrees and less than 360 degrees. some examples of deriving equivalent angles from angles that are greater than or equal to 360 degrees. angle is 360 degrees. divide it by 360 to get 1.0. subtract 1 * 360 from 360 to get 360 - 360 which gets an equivalent angle of 0 degrees. angle is 540 degrees. divide it by 360 to get 1.5. subtract 1 * 360 from 540 to get 540 - 360 which gets an equivalent angle of 180 degrees. angle is 720 degrees. divide it by 360 to get 2.0. subtract 2 * 360 from 720 to get 720 - 720 which gets an equivalent angle of 0 degrees. angle is 780 degrees. divide it by 360 to get 2.167. subtract 2 * 360 from 780 to get 780 - 720 which gets an equivalent angle of 60 degrees. angle is 3550 degrees. divide it by 360 to get 9.86111. subtract 9 * 360 from 3550 to get 3550 - 3240 which gets an equivalent angle of 310 degrees. the procedure is, once again: if you are given an angle that is greater than or equal to 360 degrees then you divide the angle by 360 degrees and then multiply 360 degrees by the integer part of the answer and then subtract the result of that from the original angle to get the equivalent angle that will be greater than or equal to 0 degrees and less than 360 degrees. IF THE ORIGINAL ANGLE IS LESS THAN OR EQUAL TO -360 DEGREES if you are given an angle that is less than or equal to -360 degrees then you divide the angle by 360 degrees and then multiply 360 degrees by the integer part of the answer and then subtract the result of that from the original angle to get the equivalent angle that will be less than or equal to 0 degrees and greater than -360 degrees. if the equivalent angle is negative, then you need to add 360 to it to make it into a positive equivalent angle. some examples of equivalent angles derived from negative angles that are less than or equal to -360 degrees: angle is -360 degrees. divide it by 360 to get -1.0. subtract -1 * 360 from 360 to get -360 - (-360) which becomes -360 + 360 which gets an equivalent angle of 0 degrees. angle is -540. divide it by 360 to get -1.5. subtract -1 * 360 from -540 to get -540 - (-360) which becomes -540 + 360 which gets a negative equivalent angle of -180 degrees. add 360 to it to get a positive equivalent angle of 180 degrees. angle is -720 degrees. divide it by 360 to get -2.0. subtract -2 * 360 from -720 to get -720 - (-720) which becomes -720 + 720 which gets an equivalent angle of 0 degrees. angle is -780 degrees. divide it by 360 to get -2.167. subtract -2 * 360 from -780 to get -780 - (-720) which becomes -780 + 720 which gets a negative equivalent angle of -60 degrees. add 360 to it to get a positive equivalent angle of 300 degrees. angle is -3550 degrees. divide it by 360 to get -9.86111. subtract -9 * 360 from -3550 to get -3550 - (-3240) which becomes -3550 + 3240 which gets a negative equivalent angle of -310 degrees. add 360 to it to get a positive equivalent angle of 50 degrees. the procedure is, once again: if you are given an angle that is less than or equal to -360 degrees then you divide the angle by 360 degrees and then multiply 360 degrees by the integer part of the answer and then subtract the result of that from the original angle to get the equivalent angle that will be less than or equal to 0 degrees and greater than -360 degrees. if the equivalent angle is negative, then you need to add 360 to it to make it into a positive equivalent angle. ANGLES BETWEEN 0 AND 360 DEGREES AND THE QUADRANT THEY ARE IN positive equivalent angles will be greater than or equal to 0 degrees and less than 360 degrees. angles between 0 and 360 degrees are divided into quadrants. angles between 0 and -360 degrees are divided into the same quadrants. the x-axis is the horizontal line on the graph. the y-axis is the vertical line on the graph. the origin is the intersection of the x-axis and the y-axis. the x-axis to the left of the origin represents 180 degrees or -180 degrees. the x-axis to the right of the origin represents 0 degrees or 360 degrees or -360 degrees. the y-axis above the origin represents 90 degrees or -270 degrees. the y-axis below the origin represents 270 degrees or -90 degrees. a positive angle rotates counter-clockwise on the graph. a negative angle rotates clockwise. the angle itself is represented by a line on the graph that starts from the origin and extends outward. here's a picture of the x and y axis values for a graph of a positive angle from 0 to 360 degrees and a graph of a negative angle from 0 to -360. the graph for the positive angle is shown on the left. the graph for the negative angle is shown in the middle. the graph for both positive angles and negative angles is shown on the right. <img src = "http://theo.x10hosting.com/2011/aug311.jpg" alt = "$$$$$" / > REFERENCE ANGLE the reference angle is a positive angle that is between 0 and 90 degrees. every angle has a reference angle associated with it. the reference angle has the same value for its trigonometric function as the angle it is being created from, except for the sign. the trigonometric functions are sine, cosine, tangent, cotangent, secant, cosecant. the sign of the trigonometric function for a reference angle is always positive. if you put the reference angle into your calculator and ask for the trigonometric function associated with it, you will always get a positive answer. this is because the reference angle is between 0 and 90 degrees and looks to the calculator as if it is in the first quadrant, even though it is not. the sign of the trigonometric function for the equivalent angle itself is dependent on the quadrant that the equivalent angle is in. FINDING THE REFERENCE ANGLE in order to find the reference angle, first find the equivalent angle. the equivalent angle is the angle that is greater than or equal to 0 degrees and less than 360 degrees. once you find the equivalent angle, you are ready to find the reference angle. RULES FOR DETERMINING THE REFERENCE ANGLE FROM THE EQUIVALENT ANGLE if the equivalent angle is between 0 and 90 degrees, then do nothing. you already have the reference angle. if the equivalent angle is between 90 degrees and 180 degrees, then the reference angle is equal to 180 degrees minus the angle. if the equivalent angle is between 180 degrees and 270 degrees, then the reference angle is equal to the angle minus 180 degrees. if the equivalent angle is between 270 and 360 degrees, then the reference angle is equal to 360 degrees minus the angle. examples: <pre> EQUIVALENT ANGLE REFERENCE ANGLE FORMULA USED 45 45 None (already have reference angle) 135 45 180 - 135 = 45 (angle is in quadrant 2) 225 45 225 - 180 = 45 (angle is in quadrant 3) 315 45 360 - 315 = 45 (angle is in quadrant 4) </pre> all of these angles have a reference angle of 45 degrees. this is for demonstration purposes only. the reference angle will not always be 45 degrees. here's some examples where it is not. <pre> EQUIVALENT ANGLE REFERENCE ANGLE FORMULA USED 0 0 None (already have reference angle) 90 90 None (already have reference angle) 180 0 180 - 180 = 0 (angle in quadrant 2 or 3) 270 90 270 - 180 = 0 (angle in quadrant 3 or 4) 150 30 180 - 150 = 30 (angle in quadrant 2) 237 57 237 - 180 - 57 (angle in quadrant 3) 322 38 360 - 322 = 38 (angle in quadrant 4) </pre> note that some angles are right on the border between quadrants. 90 is such an angle. 180 is such an angle. 360 is such an angle. the angle of 360 degrees gives you a reference angle of 0 degrees. the angle of 180 degrees gives you a reference angle of 0 degrees. the angle of 270 degrees give you a reference angle of 90 degrees. here's a picture of what we just did using the 45 degree reference angle as the example. . <img src = "http://theo.x10hosting.com/2011/sep011.jpg" alt = "$$$$$" / > you can see in the picture that all reference angles equal 45 degrees. they are, however, in different quadrants. all these angles will have the same value for the same trigonometric function. because they are in different quadrants, however, the sign of the trigonometric function for the equivalent angle might be different from the sign of the trigonometric function for the reference angle, depending on the quadrant that the equivalent angle is in. RULES FOR THE SIGN OF THE TRIGONOMETRIC FUNCTION DEPENDING ON THE QUADRANT THAT THE EQUIVALENT ANGLE IS IN the trigonometric functions from the reference angle are all positive. if the equivalent angle is in quadrant 1, the trigonometric functions derived from the reference angle all stay positive. sine = positive (+) cosine = positive (+) tangent = positive (+) if the equivalent angle is in quadrant 2, then the trigonometric functions derived from the reference angle become as shown below: sine = positive (+) cosine = negative (-) tangent = negative (-) if the equivalent angle is in quadrant 3, then the trigonometric function derived from the reference angle become as shown below: sine = negative (-) cosine = negative (-) tangent = positive (+) if the equivalent angle is in quadrant 4, then the trigonometric functions derived from the reference angle become as shown below: sine negative (-) cosine = positive (+) tangent = negative (-) you can see this visually by looking at the angle in each quadrant, as shown below. note that the hypotenuse is always positive. in the graph: h stands for hypotenuse of a right triangle. o stand for side opposite the desired angle in the right triangle. a stands for side adjacent to the desired angle in the right triangle. note that the opposite side is always vertical which is the same orientation as the y-axis. note also that the adjacent side is always horizontal which is the same orientation as the x-axis. the angle is always measured from the origin. one side of the angle is on the x-axis. the other side of the angle is a straight line that extends from the origin outward. the length of this line is equal to 1. this makes the hypotenuse of any triangle formed equal to 1. this makes the length of the side opposite the angle formed equal to the sine. this makes the length of the side adjacent to the angle formed equal to the cosine. for example: the angle is 30 degrees. the hypotenuse is equal to 1. the opposite side is equal to 1/2 which is the sine of 30 degrees. the adjacent side is equal to sqrt(3)/2 which is the cosine of 30 degrees. since this is a right triangle, then the hypotenuse squared is equal to the sum of each leg squared. this means that 1 squared is equal to (1/2) squared plus (sqrt(3)/2) squared. use your calculator to solve and you will see that: (1/2) squared plus (sqrt(3)/2) squared is equal to 1, as it should. the sine of the angle is equal to opposite / hypotenuse which is equal to o / h. the cosine of the angle is equal to adjacent / hypotenuse which is equal to a / h. the tangent of the angle is equal to opposite / adjacent which is equal to o / a. <img src = "http://theo.x10hosting.com/2011/sep012.jpg" alt = "$$$$$" / > this could also be summarized as follows: the sine of the angle is positive in the first and second quadrant. the sine of the angle is negative in the third and fourth quadrant. the cosine of the angle is positive in the first and fourth quadrant. the cosine of the angle is negative in the second and third quadrant. the tangent of the angle is positive in the first and third quadrant. the tangent of the angle is negative in the second and fourth quadrant. here's a table that summarizes the signs of the trigonometric funtions in each quadrant. <pre> quadrant 1 quadrant 2 quadrant 3 quadrant 4 sine + + - - --------------| cosine + - - + ---------| | tangent + - + - -----| | | | | | cotangent + - + - -----| | | secant + - - + ---------| | cosecant + + - - --------------| </pre> note that: cosecant is reciprocal of sine and carries the same sign (cosecant = 1/sine). secant is reciprocal of cosine and carries the same sign (secant = 1/cosine). cotangent is reciprocal of tangent and carries the same sign (cotangent = 1/tangent). so there you have it. before you find the reference angle, you have to find the positive equivalent angle. the positive equivalent angle is an angle that is greater than or equal to 0 degrees and less than 360 degrees. once you find the positive equivalent angle you have to find the reference angle. the reference angle is a positive angle that is between 0 and 90 degrees. once you find the reference angle, then you need to find the trigonometric function associated with the reference angle. once you find the trigonometric function associated with the reference angle, you need to change the sign of the trigonometric function depending on which quadrant the equivalent angle is in. EXAMPLE OF A PROBLEM INVOLVING THE REFERENCE ANGLE AND THE EQUIVALENT ANGLE here's a problem that was previously posted on algebra.com with its solution. you should see the concepts above at work in this problem. PROBLEM sin 2x + cos 60 = 0, 0 < x < 360. QUESTION FROM STUDENT i have been able to find out that sine inverse of -1/2 is -30, but how do i find answers in the 3rd and fourth quadrant for 2x?? then i can find x. my answers are coming wrong, and i can apply no suitable rule. please help, and show working if possible. SOLUTION your equation is: sin (2x) + cos(60) = 0 since cos(60) = 1/2, this equation becomes: sin (2x) + 1/2 = 0 subtract 1/2 from both sides of this equation to get: sin (2x) = -(1/2) the sine is negative in the third and fourth quadrants only. since you are searching for the reference angle at this point in the problem, then solve for the angle as if it was in the first quadrant. this means that the sign of the trigonometric function will be positive. your equation of sin(2x) = -1/2 becomes sin(2x) = 1/2 this means that 2x = sin^-1(1/2) = arcsin(1/2) which becomes 2x = 30 degrees. your answer would be 2x = 30 degrees if the angle was in the first quadrant. 2x = 30 degrees is your reference angle. since the sine of the angle you are looking for is negative, then your angle will be in the third and fourth quadrants. these angles will be your equivalent angles. the equivalent angle in the third quadrant would be 180 + 2x which becomes 180 + 30 which would make that angle equal to 210 degrees. the equivalent angle in the fourth quadrant would be 360 - 2x which becomes 360 - 30 which would make that angle equal to 330 degrees. substitute these values for the angle in your original equation and you get: sin(210) + cos(60) = 0 sin(330) + cos(60) = 0 solve both of these equations using your calculator and you'll see that they are both true. sin(210) = -1/2 cos(60) = 1/2 sin(330) = -1/2 a picture of what we just did is shown below. <img src = "http://theo.x10hosting.com/2011/aug302.jpg" alt = "$$$$$"/ > we first solved for the reference angle by making the sign of the trigonometric function positive. that gave us an angle that is in the first quadrant. we then determined what quadrant the angle had to be in if the sign of the trigonometric function was negative. in this case, the trigonometric function was the sine. when the sine is negative, the angle has to be in the third or fourth quadrant. we then solved for the equivalent angle in the third and fourth quadrant by using the following rules. if the equivalent angle is in the third quadrant, then the angle is equal to 180 plus the reference angle. if the equivalent angle is in the fourth quadrant, then the angle is equal to 360 minus the reference angle. we found the equivalent angle in the third and fourth quadrant because the sine is negative in the third and fourth quadrant. the angles we found were 210 degrees and 330 degrees. note that the angles were: 2x = 210 degrees 2x = 330 degrees now that you are done, you can convert the answer to x. you get: x = 210/2 = 105 degrees. x = 330/2 = 165 degrees. that would be your answer if you were looking for x. you do not want to convert 2x to x at an intermediate point in the solution. that could lead to difficulties. wait until the problem is solved and then convert. A WORD ABOUT CALCULATORS if you look for the sine of 210 degrees in your calculator, your calculator will tell you that the sine of 210 degrees is equal to - 1/2. so far so good. the problem comes in when you when you look for the angle whose sine is -1/2. the calculator will give you an answer that will be correct for one of your solutions, but not necessarily correct for all of your possible solutions. this answer will also probably not be in the form you want it to be in. in this case, my calculator told me that the angle whose sine is equal to -1/2 is -30 degrees. this is technically correct but not in the form i needed it to be in. it also did not provide me with the complete answer. to get the answer in the form that i needed it to be in, i assumed that the sine was equal to 1/2 rather than -1/2. that allowed me to find the reference angle from the sine. once i found the reference angle, i was able to find the equivalent angle by knowing that an angle with a negative sine had to be in the third or fourth quadrant only.
