Given cos 67.5° = [√(2+√2)]/2, find tan 67.5° , simplify where needed, and show work.
I'm starting to learn this stuff, and I'm so confused where to start. I know they gave me the coordinate X as in cos 67.5° = [√(2+√2)]/2, and I also know that tan = sin/cos, but in order for me to get there, I'll need to know sin first right? Please help me.
you know two sides of the triangle..
lets work on that cosine
(2+sqrt2)^2=4+2sqrt2+2=6+2sqrt2
the third side of triangle..
sqrt((2+sqrt2)^2 +2^2)
sqrt(10+2sqrt2)
check that. If correct, then
sine 67.5=(10+2sqrt2)/2
First of all, you have a typo:
cos 67.5 = √(2-√2) /2
Have you come across the identity
(sinØ)^2 + (cosØ)^2 = 1 ?
We can apply it here:
(sin67.5)^2 + (2 - √2)/4 = 1
sin67.5 ^2 = 1 - (2 - √2)/4
= ( 4 - 2 + √2)/4
= (2 + √2)/4
sin67.5 = [√(2+√2)]/2
so tan67.5
= sin67.5/cos67.5
= [√(2+√2)]/2 / [√(2-√2)]/2
= √(2+√2) / √(2-√2)
or
= √[ (2+√2)/(2-√2) ]
take over
Sure! Let's break it down step by step.
We are given the value of cos 67.5°, which is [√(2+√2)]/2.
To find tan 67.5°, we need to first find sin 67.5°, since tan = sin/cos.
To find sin 67.5°, we can use the trigonometric identity sin 2θ = 2sinθcosθ. In this case, we can rewrite sin 67.5° as sin (2 * 33.75°) because we know that 67.5° is the sum of two 33.75° angles.
Let's find sin (2 * 33.75°):
sin (2 * 33.75°) = 2sin(33.75°)cos(33.75°)
Now, we need to find the values of sin(33.75°) and cos(33.75°).
To do that, we can use the half-angle formulas:
sin(θ/2) = √((1 - cosθ) / 2)
cos(θ/2) = √((1 + cosθ) / 2)
In this case, θ = 67.5°, so we can write:
sin(33.75°) = √((1 - cos(67.5°))/2)
cos(33.75°) = √((1 + cos(67.5°))/2)
Now, we substitute the given value of cos(67.5°), which is [√(2+√2)]/2:
sin(33.75°) = √((1 - [√(2+√2)]/2 )/2)
cos(33.75°) = √((1 + [√(2+√2)]/2)/2)
Now we can substitute these values into our initial equation:
sin (2 * 33.75°) = 2√((1 - [√(2+√2)]/2 )/2) * √((1 + [√(2+√2)]/2)/2)
Simplifying further:
sin (2 * 33.75°) = √(2 - √2)/2 * √(2 + √2)/2
sin (2 * 33.75°) = (2 - √2) / 2 * (2 + √2) / 2
sin (2 * 33.75°) = (4 - 2√2 + 2√2 - 2) / 4
sin (2 * 33.75°) = (2 - 2) / 4
sin (2 * 33.75°) = 0
Since sin 67.5° = sin (2 * 33.75°), we have found that sin 67.5° = 0.
Now, we can find tan 67.5° using the formula tan = sin/cos:
tan 67.5° = sin 67.5° / cos 67.5°
Substituting the values we found:
tan 67.5° = 0 / [√(2+√2)]/2
tan 67.5° = 0 * 2 / [√(2+√2)]
tan 67.5° = 0
So, the simplified value of tan 67.5° is 0.