If sin = k, find tan . A. k/sqrt(1-k2) B. sqrt(1-k2)/k C. 1/sqrt(1-k2) D. k/sqrt(k2-1)

Here are the solutions to the questions:

Question 40: If $\sin \theta = k$, find $\tan \theta$.

Step 1: Use the Pythagorean identity to find $\cos \theta$.
We know that $\sin^2 \theta + \cos^2 \theta = 1$.
Substituting $\sin \theta = k$:

$$k^2 + \cos^2 \theta = 1$$ $$\cos^2 \theta = 1 - k^2$$ $$\cos \theta = \sqrt{1 - k^2}$$

Step 2: Use the definition of $\tan \theta$.

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Substitute the expressions for $\sin \theta$ and $\cos \theta$:

$$\tan \theta = \frac{k}{\sqrt{1 - k^2}}$$

The correct option is A.

Question 41: Evaluate $101.5^2 - 100.5^2$.

Step 1: Apply the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$.
Here, $a = 101.5$ and $b = 100.5$.

$$101.5^2 - 100.5^2 = (101.5 - 100.5)(101.5 + 100.5)$$

Step 2: Perform the arithmetic.

$$(101.5 - 100.5) = 1$$ $$$$