Question

How do you find the exact value of \[cos58\] using the sum and difference, double angle or half angle formulas?

Answer 1

Hint: First, we need to analyze the data so that we can solve the problem. Here we are asked to find the exact value of \[cos58\] . Since we are asked to find the exact value, we need to use the sum and difference, double angle, or half-angle formulae. The exact value is the value that we cannot estimate accurately.

Complete answer:
It's the nth Chebyshev Polynomial of the first sort and \[Tn(x)\] is one of the roots of \[T44(x) = - T46(x)\]. One of the forty-six roots of is:
Explanation:
$58^\circ $ is not a multiple of $3^\circ $. Multiples \[1^\circ \] that are not multiples of \[3^\circ \] are not constructible using a straightedge and compass, and their trig functions are not the result of some integer composition including addition, subtraction, multiplication, division, and square roots.
That doesn't mean we can't record an expression \[58^\circ \]. Let us interpret the degree sign as a fact \[\dfrac{{2\pi }}{{360}}\]
\[e{i^{{{58}^\circ }}} = cos58^\circ + i\;sin58^\circ \]
\[e{i^{ - {{58}^\circ }}} = cos58^\circ - i\;sin58^\circ \]
$({e^{i58^\circ }} + {e^{ - i58^\circ }})= 2 \times {\cos 58^\circ }$
$\cos 58^\circ = \dfrac{1}{2}({e^{i58^\circ }} + {e^{ - i58^\circ }})$
That's not very helpful.
We can try to write out a polynomial equation with \[cos58^\circ \] as one of its roots, but it will most likely be too large to fit.
$\theta = 2^\circ $ is $180$ the of the circle. Because, $\cos 88^\circ = - \cos 92^\circ $ it means, $\cos 2^\circ $ satisfies
$\cos (44\theta ) = - \cos (46\theta )$
$\cos (180^\circ - 44\theta ) = \cos (46\theta )$
Let us solve $\theta $ it first. \[\;cosx = cosa\] has root \[x = \pm a + 360^\circ k\]
$k$ referred to an integer.
\[180^\circ - 46\theta = \pm 44\theta - 360^\circ k\]
\[46\theta \pm 44\theta = 180^\circ + 360^\circ k\]
\[\theta = 2^\circ + 4^\circ k\]
Or, \[\theta = 90^\circ + 180^\circ k\]
That's a lot of roots, and we see \[\cos = 58^\circ \] within it.
The polynomials \[Tn(x)\], also known as the Chebyshev polynomials First-order polynomials fulfill \[cos(n\theta ) = Tn(cos\theta )\] and have integer coefficients. The first few are known from the double and triple angle formulas:
\[cos(0\theta ) = 1\;\;\]So, \[T0(x) = 1\]
\[cos(1\theta ) = cos\theta \;\] So, \[T1(x) = x\]
\[cos(2\theta ) = 2co{s^2}\theta - 1\;\] So, \[\;T2(x) = 2{x^2} - 1\]
\[cos(3\theta ) = 4co{s^3}\theta - 3cos\theta \] So, \[\;T3(x) = 4{x^4} - 3x\]
We can verify a suitable recursion relation:
\[{T_{n + 1}}(x) = 2x{T_n}(x) - {T_{n - 1}}(x)\]
So, in theory, we can generate these for as large a \[n\] as we want.
If we let \[x = cos\], then the equation becomes
\[cos(44\theta ) = - cos(46\theta )\]
Gets, \[{T_{44}}(x) = - {T_{46}}(x)\]
I will solve the equation just to see how the math works:
\[8796093022208{x^{44}}\]\[ - 96757023244288{x^{42}}\]\[ + 495879744126976{x^{40}}\]\[ - 1572301627719680{x^{38}}\]\[ + 3454150138396672{x^{36}}\]\[ - 5579780992794624{x^{34}}\]\[ + 6864598984556544{x^{32}}\]\[ - 6573052309536768{x^{30}}\]\[ + 4964023879598080{x^{28}}\]

Note:
There are a lot of formulas available to solve the trigonometric problems. Generally, we use the double angle formula to verify the given identity and we prefer the double angle formula to find the exact value of the trigonometric functions. Also, we apply the reduction formula to simplify the given trigonometric expression.