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First two frames use definition of cos and sin (circle is unit circle) 3rd frame: opposite angles are equal, Angles of a triangle add up to 180 therefore top angle of upper right triangle is Beta 4th frame: Erect a perpendicular off of sin(alpha) line - this makes a small right triangle whose right most angle is Beta (because this angle added to the angle opposite beta = 90.
| cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
and sin(a+b) = sin(a) cos(b) + sin (b) cos(a) |
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| Step 2: Definition of sine and cosine |
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| Step 3: Opposite angles are equal, triangle angles sum to 180 degrees |
Step 4: Erect a perpendicular from the vertical line. |
| Step 5 (to the right): From step 4 see the upper tirangle whose hypotenuse is sin(alpha) The side opposite angle beta is hypotenuse * sin(beta) or sin(alpha) * sin(beta) The leg adjacent to beta of the same triangle is hypotenuse * cos(beta) or sin(alpha) * cos(beta) - Then the triangle at the bottom with purple and orange legs has hypotenuse of cos(alpha). It's legs are cos(alpha) * cos(beta) (purple leg) and cos(alpha) * sin(beta) (orange leg). - Adding the red and orange lines gives the vertical distance and subtracting the green from the purple line gives the horizontal distance. |
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