Question 1137376
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As you begin studying trigonometry, you need to be able to quickly find the values of the trig functions for these special angles.  Rote memorization is not a particularly good method; having a picture in your mind of where those angles comes from is a much more reliable method.<br>
For the 45 degree angle, think of the diagonal of a square.  If the side lengths of the square are 1, then the diagonal (by the Pythagorean Theorem) is sqrt(2).  Then<br>
sin(45) = cos(45) = {{{1/sqrt(2) = sqrt(2)/2}}}
tan(45) = {{{sin(45)/cos(45) = 1}}}
cot(45) = {{{1/tan(45) = 1/1 = 1}}}
sec(45) = {{{1/cos(45) = sqrt(2)}}}
csc(45) = {{{1/sin(45) = sqrt(2)}}}<br>
For the 30 and 60 degree angles, think of an equilateral triangle cut in half; that forms two 30-60-90 right triangles.  If the side length of the triangle is 1, then the short side of each 30-60-90 right triangle is 1/2 (half the length of a side of the triangle).  Then the Pythagorean Theorem give us the length of the long leg of each 30-60-90 right triangle as sqrt(3)/2.  Then<br>
sin(30) = cos(60) = {{{(1/2)/1 = 1/2}}}
cos(30) = sin(60) = {{{(sqrt(3)/2)/1 = sqrt(3)/2}}}
tan(30) = cot(60) = {{{(1/2)/(sqrt(3)/2) = 1/sqrt(3) = sqrt(3)/3}}}
csc(30) = sec(60) = {{{1/(1/2) = 2}}}
csc(60) = sec(30) = {{{1/(sqrt(3)/2) = 2/sqrt(3)}}}
cot(30) = tan(60) = {{{1/(1/sqrt(3)) = sqrt(3)}}}<br>