Trigonometry

Step 1: Finding the Normal Vector to the Plane
The given vectors lying in the plane are:
The normal vector to the plane is given by the cross product:
Computing the determinant:
Expanding along the first row:
Evaluating the determinants:
Thus, the normal vector is:

Step 2: Equation of the Plane
The plane equation is given by:
Simplifying,

Step 3: Finding Perpendicular Distance from Origin
The perpendicular distance from the origin to the plane is given by:

Step 4: Finding
Thus, the final answer is:

Step 1: Express the Given Equation in Standard Form
We are given: Comparing with the standard form , we first determine :

Step 2: Express Trigonometric Components
Let: Dividing both by 17, Rewriting the equation: Using the cosine addition formula:

Step 3: Finding the Required Expression
We need to evaluate: Using the same transformation: where and , giving: Multiplying by 17:

We begin by solving the given trigonometric equations:

Step 1: Find the solution set for
Since: The general solution for is: Thus, the solution set is:

Step 2: Find the solution set for
We are given: Multiplying both sides by : Solving for : Since: we substitute: which corresponds to: Thus, the solution set is:

Step 3: Compute
Comparing the solution sets, we find: Thus, the correct answer is:

Step 1: Using Sum-to-Product Identities
We use the sum-to-product formulas:

Step 2: Expressing in Terms of Known Values
Given: From the sum-to-product identities:

Step 3: Squaring and Summing
Squaring both equations and summing: Since ,

Step 4: Solving for
Taking square roots: Since lies in the 3rd or 4th quadrant, we take the negative value:

We are given the conditions: where: - are the exradii, - is the circumradius of .

Step 1: Recall the exradius and circumradius relation
For any triangle, From the given conditions, Adding both equations, Since we know , subtracting gives: Substituting in : Similarly, from :

Step 2: Identify the type of triangle
Since , , and , we use the identity: For : which satisfies the given conditions. This confirms that , and the triangle is right-angled but non-isosceles. Thus, the correct answer is:

We need to evaluate:

Step 1: Express in terms of
Since we rewrite the given expression as:

Step 2: Expand the expression
Distribute : Using the identity: we obtain:

Step 3: Substitute values of and
From trigonometric tables: Substituting these: Since: we conclude: Thus, the correct answer is .

We are given the equation:

Step 1: Define the range of inverse sine function
For , we know: Similarly, since is also defined in the range: Multiplying both sides of this inequality by 2: Since the principal range of is limited to , for the given equation to have a valid solution, we must satisfy:

Step 2: Solve for
Dividing the inequality by 2: Taking sine on both sides: Since , we obtain:

Step 3: Conclusion
Thus, the equation has a solution only when: Hence, the correct answer is:

We need to determine , the exradius corresponding to side , for an isosceles triangle with base .

Step 1: Recall the formula for the exradius
The exradius corresponding to a side of a triangle is given by: where: - is the area of the triangle, - is the semi-perimeter, - is the length of the side opposite to the exradius.

Step 2: Express in terms of circumradius
For a triangle, Since the triangle is isosceles, angles and are equal: Using the identity: we get:

Step 3: Compute
Since is given by: Thus, the correct answer is:

We are given:

Step 1: Compute using the identity
Using the hyperbolic identity: Substituting :

Step 2: Compute and using triple angle formulas
The standard identities for triple angles are: Substituting the values: Similarly,

Step 3: Compute
Thus, the correct answer is:

Step 1: Use the Law of Sines
In a triangle, Thus, Differentiating both sides with respect to ,

Step 2: Compute the Error Propagation
For small errors, the approximate error in due to an error in is: Dividing by ,

Step 3: Convert to Percentage Error
Percentage error in : Since , and using , we get:

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Understanding the Given Integrals
We analyze the given integral equations and their recurrence relations: Using reduction formulas for trigonometric integrals: Similarly, This provides relationships between the powers of sine in the integrals.

Step 2: Deriving the Relationship
Given: Using recurrence properties, Rearranging,

Step 3: Conclusion
Thus, the correct answer is:

Step 1: Consider the Given Integral
We need to evaluate the integral: Using the identity: we rewrite: Let: Using the Weierstrass substitution: Rewriting in terms of :

Step 2: Evaluating at
Using , Since: Thus,

Step 3: Compute
Since , we get:

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Analyze Assertion (A):

Let , , . Let . Then , etc. Consider . Since the product , the sum lands in the second quadrant (or similar), so we use the formula or logic involving sums of angles being . Specifically, it can be shown that , which implies . Since terms are positive, sum is . Assertion (A) is True.
Step 2: Analyze Reason (R):

The formula is the standard addition theorem used to prove such identities (often applied sequentially). It is true for . If , it is . However, technically, R states the formula for . While the strict formula for has a shift, the fundamental algebraic manipulation relies on the tangent addition formula provided in R. In the context of exam logic, this formula is the basis for proving (A).
Step 4: Final Answer:

Both are true and R explains A.