Answer to Question #250578 in Trigonometry for umar

Question

Determine the numerical value of the following expression without the use of a calculator:

 

log10 (1000100)

100

+

X100

n=1

sin(n) + 1

(􀀀1)n

!

vuut

1Y000

m=1

1

cos(m)2

Answer 1

Question: Determine the numerical value of the following expression without the use of a calculator:

$$(\frac{\log_{10} (1000^{100})} {100} + \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}) . \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}$$

Solution:

For the logarithm operation:

$$\frac{\log_{10} (1000^{100})} {100}$$

$= \frac{100\log_{10} 1000} {100}$

$= \frac{100\log_{10} 10^3} {100}$

$= \frac{(3\times 100)\log_{10} 10} {100}$

since $\log_{10} 10 = 1$;

Then,

$$\therefore\ \frac{\log_{10} (1000^{100})} {100} = \frac{(3\times 100)\log_{10} 10} {100} = \frac{(3\times 100 \times 1)} {100} = 3$$

For the cycle operations:

$$f(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}$$

$for \ all \ {n \in \mathbb{N}}, \sin(\pi n) = 0 \\ considering \ the \ denominator, \\ for \ even \ n, \frac{\sin(\pi n) + 1} {(-1)^n} =1 \\ for \ odd \ n, \frac{\sin(\pi n) + 1} {(-1)^n} = -1$

Therefore, dividing the summation into two parts: i.e. 50 odds and 50 evens, we have:

$f(n) = \\ 50 \times (\frac{\sin(\pi n) + 1} {(-1)^n}) \ for\ denominator\ n = evens \\ + 50 \times (\frac{\sin(\pi n) + 1} {(-1)^n}) \ for\ denominator\ n = odds \\\ = (50 \times 1) + (50\times -1) = 50 - 50 = 0$

$$\therefore \ f(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n} = 0$$

Also, for:

$$f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}$$

$for \ all \ {m \in \mathbb{N}}, \frac{1} {\cos(\pi m)^2} = 1$

Therefore the product of:

$$\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}} = 1 \times 1 \times 1 \times........ = 1 \\ \\ \therefore \ the \ square \ root \ of \ the \ product \ of: \\ \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} = \sqrt{1} = 1$$

$$\therefore \ f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} = 1$$

Therefore, the final answer to the expression is given by:

$$\therefore (\frac{\log_{10} (1000^{100})} {100} + \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}) . \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}$$

$$= (3 + 0) \times 1$$ $$= 3 \times 1$$ $$= 3$$