Cos^2(π/4): Why Does it Equal 1/2?

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In summary, cos^2(π/4) is equal to 1/2, which can also be written as 0.5 or 50%. This is because in a 45-45-90 triangle, the cosine of π/4 is equal to 1/√2, which squared gives us 1/2. This can be proven using the Pythagorean theorem. While cos^2(π/4) is always equal to 1/2 in a 45-45-90 triangle, it may have different values in other types of triangles. Additionally, cos^2(π/4) is related to the sine and tangent functions through the Pythagorean identity.
  • #1
CrossFit415
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cos^2(∏/4)

Why does this equal to 1/2? Doesn't Cos(Pi/4)= √2/2 ?

Thanks
 
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  • #2
What is (√2/2)^2?
 
  • #3
[itex]\cos^2 \frac{\pi}{4}[/itex]
[itex]= \left(\cos \frac{\pi}{4}\right)\left(\cos \frac{\pi}{4}\right)[/itex]
[itex]= \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)[/itex]
[itex]= ?[/itex]
 
  • #4
Opps! Haha, thank you both, I know understand cos pi 4 now.
 
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FAQ: Cos^2(π/4): Why Does it Equal 1/2?

What is the value of cos^2(π/4)?

The value of cos^2(π/4) is equal to 1/2. This can also be written as 0.5 or 50%.

Why does cos^2(π/4) equal 1/2?

This is because the cosine function is equal to the adjacent side over the hypotenuse in a right triangle. In a 45-45-90 triangle, the adjacent and opposite sides are equal, so the cosine of π/4 (45 degrees) is equal to 1/√2. Squaring this value gives us 1/2.

Can you prove that cos^2(π/4) equals 1/2?

Yes, using the Pythagorean theorem, we can see that in a 45-45-90 triangle, the hypotenuse (c) is equal to the square root of 2 times one of the sides (a or b). Therefore, the cosine of π/4 is equal to a/c, which is equal to 1/√2. Squaring this value gives us 1/2.

Is cos^2(π/4) always equal to 1/2?

Yes, in a 45-45-90 triangle, the cosine of π/4 will always be equal to 1/2. However, in other types of triangles, the value of cos^2(π/4) may be different.

How is cos^2(π/4) related to other trigonometric functions?

Cos^2(π/4) is equal to 1/2, which is also equal to the square of the sine function at π/4 or the tangent function at π/4. In general, the cosine function is related to the sine and tangent functions through the Pythagorean identity: cos^2(x) + sin^2(x) = 1.

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