6) The inverse relation of the quadratic equation f(x) = (1 / 3) · x² - 3 · x + 5 is x = 4.5 ± √(3 · f(x) + 5.25).
7) The sum of the arithmetic series is s = ∑ [5 + 13 · (n - 1)], for n ∈ {1, 2, 3, ..., n}.
8) The common ratio of the geometric sequence is 2.
9) The exact value of sec A is - 13 / 12.
How to find the inverse of quadratic function and arithmetic sums and geometric sequences and trigonometric functions
6) In this problem we need to find the inverse relation of a quadratic equation by algebraic measures:
f(x) = (1 / 3) · x² - 3 · x + 5
3 · f(x) = x² - 9 · x + 15
3 · f(x) + 5.25 = x² - 9 · x + 20.25
3 · f(x) + 5.25 = (x - 4.5)²
± √(3 · f(x) + 5.25) = x - 4.5
x = 4.5 ± √(3 · f(x) + 5.25)
The inverse relation of the quadratic equation f(x) = (1 / 3) · x² - 3 · x + 5 is x = 4.5 ± √(3 · f(x) + 5.25).
7) According to the definition of the arithmetic sum, the sum is represented by the formula:
s = ∑ [a + r · (n - 1)], for n ∈ {1, 2, 3, ..., n} (1)
Where:
- a - First term of the series.
- r - Change between two consecutive elements of the series.
If we kwow that a = 5 and r = 13, then the sum of the arithmetic series is:
s = ∑ [5 + 13 · (n - 1)], for n ∈ {1, 2, 3, ..., n}
The sum of the arithmetic series is s = ∑ [5 + 13 · (n - 1)], for n ∈ {1, 2, 3, ..., n}.
8) Geometric sequences are generated by the following expression:
s = a · rⁿ ⁻ ¹, for n ∈ {1, 2, 3, ..., n}
Where:
- a - First element of the series.
- r - Common ratio
In accordance with the statement, we find that:
32 = r¹⁰ / r⁵
32 = r⁵
r = 2
The common ratio of the geometric sequence is 2.
9) The exact value of the secant function can be found by means of this trigonometric expression:
sec A = 1 / cos A
sec A = 1 / (- 12 / 13)
sec A = - 13 / 12
The exact value of sec A is - 13 / 12.
Remark
Exercise 9 is incomplete and poorly formatted. Correct form is shown below:
The angle A is found in quadrant III, such that cos A = - 12 / 13. Determine the exact value of sec A.
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