using Test

# Part 2.1

function sin(x)
    soma = big(0)
    for i in 0:10
        n = 1 + (i * 2)
        soma += (-1)^i * (x^n) / factorial(big(n))
    end
    return soma
end
  
function cos(x)
    soma = big(0)
    for i in 0:10
        n = i * 2
        soma += (-1)^i * big(x^n) / factorial(big(n))
    end
    return soma
end

function tan(x)
    soma = big(0)
    for n in 1:10
        soma += (2^(2 * n) * (2^(2 * n) - 1) * bernoulli(n) * x^(2 * n - 1)) / factorial(2 * n)
    end
    return soma
end

function bernoulli(n)
    n += 1
    A = Vector{Float64}(undef, n + 1)
    for m = 0 : n
        A[m + 1] = 1 // (m + 1)
        for j = m : -1 : 1
            A[j] = j * (A[j] - A[j + 1])
        end
    end
    return A[1]
end

# ---

function quaseigual(v1, v2)
    erro = 0.0001
    igual = abs(v1 - v2)
    if igual <= erro
        return true
    else
        return false
    end
end

function check_sin(value, x)
    return quaseigual(value, Base.sin(x))
end

function check_cos(value, x)
    return quaseigual(value, Base.cos(x))
end

function check_tan(value, x)
    return quaseigual(value, Base.tan(x))
end

# ---

function taylor_sin(x)
    soma = big(0)
    for i in 0:10
      n = 1 + (i * 2)
      soma += (-1)^i * (x^n) / factorial(big(n))
    end
    return soma
  end
  
function taylor_cos(x)
    soma = big(0)
    for i in 0:10
        n = i * 2
        soma += (-1)^i * big(x^n) / factorial(big(n))
    end
    return soma
end

function taylor_tan(x)
    soma = big(0)
    for n in 1:10
        soma += (2^(2 * n) * (2^(2 * n) - 1) * bernoulli(n) * x^(2 * n - 1)) / factorial(2 * n)
    end
    return soma
end

function bernoulli(n)
    n *= 2
    A = Vector{Rational{BigInt}}(undef, n + 1)
    for m = 0 : n
        A[m + 1] = 1 // (m + 1)
        for j = m : -1 : 1
            A[j] = j * (A[j] - A[j + 1])
        end
    end
    return abs(A[1])
end