Wie bewerten Sie #cos (pi / 8) #?

Antworten:

#cos(pi/8) = sqrt(1/2+sqrt(2)/4)#

Erläuterung:

#"Use the double-angle formula for cos(x) : "#
#cos(2x) = 2 cos^2(x) - 1#
#=> cos(x) = pm sqrt((1 + cos(2x))/2)#
#"Now fill in x = "pi/8#
#=> cos(pi/8) = pm sqrt((1 + cos(pi/4))/2)#
#=> cos(pi/8) = sqrt((1+sqrt(2)/2)/2)#
#=> cos(pi/8) = sqrt(1/2+sqrt(2)/4)#

#"Remarks : "#
#"1) "cos(pi/4) = sin(pi/4) = sqrt(2)/2" is a known value"#
#"because "sin(x) = cos(pi/2-x)," so "#
#sin(pi/4)=cos(pi/4)" and "sin^2(x)+cos^2(x) = 1#
#=> 2 cos^2(pi/4) = 1 => cos(pi/4) = 1/sqrt(2) = sqrt(2)/2.#
#"2) because "pi/8" lies in the first quadrant, "cos(pi/8) > 0", so"#
#"we need to take the solution with the + sign."#

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