Evaluate (12.76 sin 33.45° / (9.063)²) ^ (1/3) using logarithms.

Here are the solutions to questions 3 and 4:

Question 3:
To evaluate $\left(\frac{12.76 \sin 33.45^\circ}{(9.063)^2}\right)^{\frac{1}{3}}$ using logarithms:

Let $N = \left(\frac{12.76 \sin 33.45^\circ}{(9.063)^2}\right)^{\frac{1}{3}}$.
Then $\log N = \frac{1}{3} [\log(12.76) + \log(\sin 33.45^\circ) - 2\log(9.063)]$.

Step 1: Find the value of $\sin 33.45^\circ$.
From sine tables, $\sin 33.45^\circ = 0.5512$.

Step 2: Find the logarithms of the numbers.

• $\log 12.76 = 1.1058$
• $\log (\sin 33.45^\circ) = \log 0.5512 = \bar{1}.7413$
• $\log 9.063 = 0.9572$

Step 3: Calculate the logarithm of the numerator.

$$\log(12.76) + \log(\sin 33.45^\circ) = 1.1058 + \bar{1}.7413$$ $$= 1.1058 + (-1 + 0.7413)$$ $$= 1.1058 - 0.2587 = 0.8471$$

Step 4: Calculate the logarithm of the denominator.

$$2 \log(9.063) = 2 \times 0.9572 = 1.9144$$

Step 5: Subtract the logarithm of the denominator from the logarithm of the numerator.

$$\log\left(\frac{12.76 \sin 33.45^\circ}{(9.063)^2}\right) = 0.8471 - 1.9144 = -1.0673$$

Step 6: Divide by 3 to find $\log N$.

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