# Lect4A - Triangle Definition of sin and cos Consider the...

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92.131 Lecture 4A 1 of 25 Ronald Brent © 2009 All rights reserved. Triangle Definition of sin θ and cos Consider the triangle ABC below. Let A be called . ADJ (side adjacent to the angle θ ) OPP (side opposite to the angle ) HYP (hypotenuse) A B C Then B A C B HYP OPP sin = = B A C A HYP ADJ cos = = . (SOH CAH TOA)

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92.131 Lecture 4A 2 of 25 Ronald Brent © 2009 All rights reserved. 1 45 o 45 o s s 1 45 o 45 o 1 2 1 2 Special Angles (30 ° , 45 ° , 60 ° ) Find o 45 sin and o 45 cos . Pythagorean Theorem: 1 2 2 = + s s , or 1 2 2 = s Hence 2 1 and 2 1 2 ± = = s s . Choosing s > 0, we have 2 1 = s , and so 2 2 2 1 45 cos 45 sin = = = o o
92.131 Lecture 4A 3 of 25 Ronald Brent © 2009 All rights reserved. Example: Find o 30 sin , o 30 cos , o 60 sin , and o 60 cos . Begin with a o o o 90 60 30 right triangle with HYP = 1. 1 30 o 30 o 60 o 60 o s t 1 Notice that when flipping the triangle down and consider the larger one, the result is an equilateral triangle, so the vertical side is also 1. This means 2 t = 1, or 2 1 = t . Now since st 22 1 += , we have s 2 1 4 1 ,

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92.131 Lecture 4A 4 of 25 Ronald Brent © 2009 All rights reserved. 1 1 2 3 2 30 o 60 o which means 4 3 2 = s 2 3 ± = s Again, it is clear that s > 0, which implies the triangle looks like: So 2 1 30 sin = o 2 3 30 cos = o , while 2 1 60 cos = o 2 3 60 sin = o .
92.131 Lecture 4A 5 of 25 Ronald Brent © 2009 All rights reserved. Angles in Radian Measure How big is a radian? Here’s how big: it’s the angle corresponding to an arc length of 1 in a unit circle. Look at the diagram below. A “unit circle” indicates that the radius = 1 unit, and we’ll always put the center at (0,0) for convenience. The angle θ as drawn is 1 radian, because the arc length “subtended” (cut off) by the angle has length = 1 unit. (0,0) (1,0) 1 unit Length of the arc = 1 unit θ = 1 radian

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Lect4A - Triangle Definition of sin and cos Consider the...

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