What is the exact value for cos(pie/12)?

Question

Can some one help me solve this question with step by step instructions please?

Sorry Pi
I still am not getting the answer.

Cos(pi/12) = Cos(pi/4) - Cos(pi/6)
Where Cos(pi/4) = sqrt2/2 and Cos(pi/6) = sqrt3/2

Answer 1

You did ask for the EXACT value, which no one has posted yet. The exact value is √(2 + √3)/2. To three decimal places, that is .966.

As for how I got that, I started with a 30-60-90 triangle and extended the longer side by the length of the hypotenuse to get a 15-75-90 triangle, then used Pythagoragus. For some reason, though, most students have an easier time using a half-angle formula, so feel free to do it that way.

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OK, sorry, I see Awesome Sauce also gives an exact value. It is also correct, just looks a little different.

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From your comment in Additional Details, it appears you want to use the angle difference relation: cos(π/12) = cos(π/4 - π/6). That works also, but you need to get the formula right, which is:

cos(a - b) = cos a * cos b + sin a * sin b

For cos (45 - 30), this gives cos 45 * cos 30 + sin 45 * sin 30 = √2/2 * √3/2 + √2/2 * 1/2 = √2/4(√3 + 1). This is yet another expression for the same exact value.

Answer 2

I imagine the question is asking for the exact value in the form of roots, not a rounded number (which is inherently inexact). The easiest way ro remember the exact formula for cos(pi/12) radians, or cos(15) degrees is the Ailles Rectangle (see linked image).

From the two 45/90 triangles, it is easy to work out that x is √3/√2 and y is 1/√2. With this, you can use the cosine rule (adj/hyp) to get the exact value. In this case, x + y / 2, which is

(√3/√2 + 1/√2) / 2

Multiply the top through by √2/√2:

(√6/2 + √2/2) / 2

Tidy up the division to give:

√6/4 + √2/4

Answer 3

cos(PIE/12) = Lemons.

the exact value for cos(PI/12) can be found with the Half-angle trigonometric identity...

cos(theta/2) = +/-(sqrt((1+cos(theta))/2))

we know that cos(pi/6) = sqrt(3)/2

So cos(pi/6/2) = cos(pi/12)
we can drop the minus in the plus or minus, cause the value will be positive, as it is in the first quadrant.
so cos(pi/12) = (sqrt(1 + (sqrt(3)/2)/2)) This is the exact value.
Plug all that trash into the calculator and get cos(pi/12) = 0.9659... which is an approximation.

P.S. if you have to do it by half-angle identies in class, don't worry. After the final, and in Calculus, you can just use a calculator to figure it out quickly.

Answer 4

csc seventy 5° = a million/sin seventy 5° = a million/sin (30° + 40 5°) = a million/[sin 30° cos 40 5° + cos 30° sin 40 5°) = a million/(a million/2 • ?2/2 + ?3/2 • ?2/2) = a million/[(?2 + ?6)/4] = 4/(?2 + ?6) The denominator could be rationalized, yet you do no longer ought to try this.

Answer 5

cos 2 θ = 1 - 2 cos ² θ
2 cos ² θ = 1 - cos 2 θ
cos ² θ = (1/2) (1 - cos 2 θ)
cos θ = (1 / √2) (1 - cos 2 θ)^(1/2)
cos (π/12) = (1 / √2) (1 - cos π / 6) ^(1/2)
cos (π/12) = (1 / √2) (1 - √3 / 2)^(1/2)
cos (π/12) = (1 / √2) (2 - √3)^(1/2)(1 / √2)
cos (π/12) = (1/2) (2 - √3)^(1/2)

Answer 6

cos (pi/12) = cos 15 = 0.966

Answer 7

= .9659258263
Only if you use (pi) not 3.14.

P.S. its pi not pie.

.

Answer 8

0.96