Introduction To Three Dimensional Geometry

We are given that , and we are asked to find .
Step 1: From the given, , so we can write: Step 2: Using the identity , we can find : Thus, Step 3: Now, we can calculate using the identity : Step 4: Finally, we calculate the sum : To combine these, we need a common denominator: Thus, the correct answer is option (C).

We are given the expression , and we need to simplify it. Let .
Then , and the angle lies within the range . Also, recall that: Now, we can rewrite the expression as: This simplifies to: Using the trigonometric identity , we have: Since , we use the Pythagorean identity to find: Thus, the value of is .
Thus, the correct answer is option (A), .

We are given the integral: We are tasked with finding the value of .
Step 1: Use integration by parts to solve the integral. Recall the formula for integration by parts:
Let and . Then, and .
Step 2: Apply the integration by parts formula:
Simplify: The integral of is , so: Step 3: Factor out :
Thus, comparing with the given equation , we see that:

Thus, the correct answer is option (A).