TrigFunctionsofSpecialAngles

TrigFunctionsofSpecialAngles - Special Angles and their...

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Special Angles and their Trig Functions By Jeannie Taylor Through Funding Provided by a VCCS LearningWare Grant
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We will first look at the special angles called the quadrantal angles. 90 180 270 0 The quadrantal angles are those angles that lie on the axis of the Cartesian coordinate system: , , , and . 0 90 180 270
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We also need to be able to recognize these angles when they are given to us in radian measure. Look at the smallest possible positive angle in standard position, other than 0 , yet having the same terminal side as 0 . This is a 360 angle which is equivalent to . radians 2 π radians 2 360 = 90 180 270 0 2 = radians If we look at half of that angle, we have radians or 180 . radians = Looking at the angle half-way between 180 and 360 , we have 270 or radians which is of the total (360 or ). 2 3 4 3 2 radians Moving all the way around from 0 to 360 completes the circle and and gives the 360 angle which is equal to radians. 2 radians 2 3 = Looking at the angle half-way between 0 and 180 or , we have 90 or . 2
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We can count the quadrantal angles in terms of . radians 2 π radians 2 0 radians radians 2 2 radians 2 3 radians 2 4 Notice that after counting these angles based on portions of the full circle, two of these angles reduce to radians with which we are familiar, . 2 and = radians radians 2 = Add the equivalent degree measure to each of these quadrantal angles. 0 90 180 270 radians 57 . 1 radians 14 . 3 radians 71 . 4 radians 28 . 6 We can approximate the radian measure of each angle to two decimal places. One of them, you already know, . It will probably be a good idea to memorize the others. Knowing all of these numbers allows you to quickly identify the location of any angle. radians 14 . 3 360
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We can find the trigonometric functions of the quadrantal angles using this definition. We will begin with the point (1, 0) on the x axis. (1, 0) radians 2 π 0 radians radians 2 3 = radians radians 2 = 0 90 180 270 360 or As this line falls on top of the x axis, we can see that the length of r is 1. y x x y x r r x y r r y = = = = = = β cot tan sec cos csc sin For the angle 0 , we can see that x = 1 and y = 0. To visualize the length of r, think about the line of a 1 angle getting closer and closer to 0 at the point (1, 0). Remember the six trigonometric functions defined using a point ( x , y ) on the terminal side of an angle, .
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radians 2 π 0 radians radians 2 3 = radians radians 2 = 0 90 180 270 360 (1, 0) or undefined is 0 cot 0 1 0 0 tan 1 0 sec 1 0 cos undefined is 0 csc 0 0 sin = = = = = Using the values, x = 1, y = 0, and r = 1, we list the six trig functions of 0.
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TrigFunctionsofSpecialAngles - Special Angles and their...

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