What is the usual way of proving things likecos⁡(π6)=12ϕ ?

Here's a standard way of doing this with complex numbers:Let z=eπi5, which is a primitive 10th root of unity. So it's a root of the 10th cyclotomic polynomial, which is z4−z3−z2+1. In other words,e4π5−e3πi5+e2πi5−eπ5+1=0So the real part of this is also equal to zero. But the real part of this iscos⁡(4π5)−cos⁡(3π5)+cos⁡(2π5)−cos⁡(π5)+1=2cos⁡(2π5)−2cos⁡(π5)+1Let x=cos⁡(π5). Then this can be written as2(2x2−1)−2x+1=04x2−2x−1=0whose positive root is 1+54=ϕ2.

Check the solution to "What is the usual way of proving things..."

Alexandria Dodd

Answered question

2022-02-27

What is the usual way of proving things like

\cos (\frac{π}{6}) = \frac{1}{2} ϕ

Answer & Explanation

Tommie Bryan

Beginner2022-02-28Added 4 answers

Here's a standard way of doing this with complex numbers:
Let

z = e^{π \frac{i}{5}}

, which is a primitive 10th root of unity. So it's a root of the 10th cyclotomic polynomial, which is

z^{4} - z^{3} - z^{2} + 1

. In other words,

e^{4 \frac{π}{5}} - e^{3 π \frac{i}{5}} + e^{2 π \frac{i}{5}} - e^{\frac{π}{5}} + 1 = 0

So the real part of this is also equal to zero. But the real part of this is

\cos (\frac{4 π}{5}) - \cos (\frac{3 π}{5}) + \cos (\frac{2 π}{5}) - \cos (\frac{π}{5}) + 1 = 2 \cos (\frac{2 π}{5}) - 2 \cos (\frac{π}{5}) + 1

Let

x = \cos (\frac{π}{5})

. Then this can be written as

2 (2 x^{2} - 1) - 2 x + 1 = 0

4 x^{2} - 2 x - 1 = 0

whose positive root is

\frac{1 + \sqrt{5}}{4} = \frac{ϕ}{2}

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Precalculus

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix
$[\begin{matrix} 1 & 3 & 0 & - 4 \\ 2 & 6 & 0 & - 8 \end{matrix}]$
Find, correct to the nearest degree, the three angles of the triangle with the given vertices
A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)
Whether f is a function from Z to R if
a) $f (n) = \pm n$ .
b) $f (n) = \sqrt{n^{2} + 1}$ .
c) $f (n) = \frac{1}{n^{2} - 4}$ .
?
How to write the expression $6^{\frac{3}{2}}$ in radical form?
How to evaluate $\sin (\frac{- 5 π}{4})$ ?
What is the derivative of $\cot^{2} x$ ?
How to verify the identity: $\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} = 0$ ?
Find $\tan (22 . 5^{\circ})$ using the half-angle formula.
How to find the exact values of $\cos 22.5 °$ using the half-angle formula?
How to express the complex number in trigonometric form: 5-5i?
The solution set of $\tan θ = 3 \cot θ$ is
How to find the angle between the vector and $x -$ axis?
Find the probability of getting 5 Mondays in the month of february in a leap year.
How to find the inflection points for the given function $f (x) = x^{3} - 3 x^{2} + 6 x$ ?
How do I find the value of sec(3pi/4)?