If sin A = 4/5 and cos B = 5/13, then find cos(A + B). Also, find the arithmetic mean of the following discrete series using step-deviation method:

Steps to Solve

1. Find cos A Given that $ \sin A = \frac{4}{5} $, we can find $ \cos A $ using the Pythagorean identity:

$$ \cos^2 A = 1 - \sin^2 A $$

Calculating:

$$ \cos^2 A = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} $$

Thus,

$$ \cos A = \sqrt{\frac{9}{25}} = \frac{3}{5} $$

2. Find sin B Given that $ \cos B = \frac{5}{13} $, we can find $ \sin B $ using the Pythagorean identity:

$$ \sin^2 B = 1 - \cos^2 B $$

Calculating:

$$ \sin^2 B = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} $$

Thus,

$$ \sin B = \sqrt{\frac{144}{169}} = \frac{12}{13} $$

3. Apply the cos(A + B) formula We use the angle addition formula for cosine:

$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$

Substituting the values we found:

$$ \cos(A + B) = \left(\frac{3}{5}\right)\left(\frac{5}{13}\right) - \left(\frac{4}{5}\right)\left(\frac{12}{13}\right) $$

Calculating each term:

$$ \cos(A + B) = \frac{15}{65} - \frac{48}{65} = \frac{15 - 48}{65} = \frac{-33}{65} $$

4. Step-deviation method for the arithmetic mean Next, we calculate the arithmetic mean using the step-deviation method for the table provided:

5. Calculate the total values First, find the total frequency $N$:

$$ N = 2 + 3 + 4 + 2 + 1 = 12 $$

6. Find the mean Choose a suitable assumed mean, say $A = 8$.

Calculate deviations from the assumed mean and the weighted values ($d_i = x_i - A$):

• For $x=6$, $d_1 = 6 - 8 = -2$, with $f_1 = 2 \rightarrow f_1 d_1 = 2 \cdot (-2) = -4$
• For $x=7$, $d_2 = 7 - 8 = -1$, with $f_2 = 3 \rightarrow f_2 d_2 = 3 \cdot (-1) = -3$
• For $x=8$, $d_3 = 8 - 8 = 0$, with $f_3 = 4 \rightarrow f_3 d_3 = 4 \cdot 0 = 0$
• For $x=9$, $d_4 = 9 - 8 = 1$, with $f_4 = 2 \rightarrow f_4 d_4 = 2 \cdot 1 = 2$
• For $x=10$, $d_5 = 10 - 8 = 2$, with $f_5 = 1 \rightarrow f_5 d_5 = 1 \cdot 2 = 2$

Summing these values gives:

$$ \sum f_i d_i = -4 - 3 + 0 + 2 + 2 = -3 $$

Then compute the mean:

$$ \text{Mean} = A + \frac{\sum f_i d_i}{N} = 8 + \frac{-3}{12} = 8 - 0.25 = 7.75 $$