Let cos 67.5° = [√(2(+√2)]/2, find tan 67.5°. Show work and simplify.
I'm not too sure if i'm doing this correct.
I know that the given is cos 67.5° = [√(2(+√2)]/2
sin^2 x + cos^2 x = 1
x=67.5°
sin^2 67.5° + cos^2 67.5° = 1
sin^2 67.5° = 1 - ([√2(+√2)]/2)^2
sin^2 67.5° = 1 - (2+√2)/4
sin^2 67.5° = (2-√2)/4
sin 67.5° = ([√2(+√2)]/2)
and now i needa find tan.
tan 67.5° = ?
well, tan = sin/cos, right?
Also, 67.5° = 3pi/8 = tan(1/2 * 3pi/4)
so you can use the half-angle formula
tan(x/2) = sinx/(1+cosx)
tan 3pi/8 = (1/√2)/(1-1/√2)
= (1/√2) / (√2-1)/√2
= 1/(√2-1)
= √2+1
or, √((2+√2)/(2-√2))
To find tan 67.5°, you can use the relationship between sine and cosine.
We know that tan x = sin x / cos x.
Given that sin 67.5° = (√2 + √2)/2 and cos 67.5° = (√(2+√2))/2, substitute these values into the formula:
tan 67.5° = sin 67.5° / cos 67.5° = [ (√2 + √2)/2 ] / [ (√(2+√2))/2 ]
To simplify this expression, we can multiply both the numerator and the denominator by the conjugate of the denominator (√(2+√2)):
tan 67.5° = [ (√2 + √2)/2 ] * [ 2 / (√(2+√2)) ]
= [ (√2 + √2) * 2 ] / [ 2 * (√(2+√2)) ]
= [ 2√2 + 2√2 ] / [ 2√(2+√2) ]
= (4√2) / [2√(2+√2)]
Now, divide both the numerator and the denominator by 2:
tan 67.5° = 2√2 / √(2+√2)
So, the simplified value of tan 67.5° is 2√2 / √(2+√2).