Asked by Ande2
Show that Cos 22.5=√2+√2/2
Answers
Answered by
Bosnian
22.5° = 45° / 2
cos 45° = √ 2 / 2
cos ( θ / 2 ) = ± √ [ ( 1 + cos θ ) / 2 ]
45° is located in first quadrant where all trigonometric functions are positive so:
cos 22.5° = cos ( 45° / 2 ) = √ [ ( 1 + cos 45° ) / 2 ] =
√ [ ( 1 + √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 + √ 2 / 2 ) / 2 ] =
√ [ ( 2 + √ 2 ) / 2 / 2 ] = √ [ ( 2 + √ 2 ) / 4 ] =
√ ( 2 + √ 2 ) / √ 4 = √ ( 2 + √ 2 ) / 2
cos 45° = √ 2 / 2
cos ( θ / 2 ) = ± √ [ ( 1 + cos θ ) / 2 ]
45° is located in first quadrant where all trigonometric functions are positive so:
cos 22.5° = cos ( 45° / 2 ) = √ [ ( 1 + cos 45° ) / 2 ] =
√ [ ( 1 + √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 + √ 2 / 2 ) / 2 ] =
√ [ ( 2 + √ 2 ) / 2 / 2 ] = √ [ ( 2 + √ 2 ) / 4 ] =
√ ( 2 + √ 2 ) / √ 4 = √ ( 2 + √ 2 ) / 2
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