Trigonometry

Concept: Trigonometry - Properties of Triangles (Sine/Cosine Rules) and Angle Sum Property.

Step 1: Apply the angle sum property of a triangle. In any triangle, the sum of the internal angles is or radians. Therefore, . We can rewrite this to group A and C: .

Step 2: Substitute this into the given expression's angle. The given expression contains the angle term . Rearrange the numerator to group A and C together: . Substitute into this expression: .

Step 3: Simplify the trigonometric function using complementary angles. Substitute the simplified angle back into the original sine function: . Using the standard complementary angle identity , this simplifies entirely to .

Step 4: Substitute this result back into the full expression. The original expression was . Replacing the sine term with our result from Step 3, the expression becomes .

Step 5: Apply the Cosine Rule to reach the final algebraic form. The Cosine Rule for an angle in a triangle states that . We substitute this formula directly into our current expression: .
The terms in the numerator and denominator perfectly cancel each other out, leaving only the numerator of the Cosine Rule formula: . Rearranging the terms algebraically to match the options gives . $ $

Concept: Trigonometry - Sequences and Series (Harmonic Progression) and Trigonometric Identities.

Step 1: Convert the H.P. condition to an A.P. condition. If three terms are in Harmonic Progression (H.P.), their reciprocals must be in Arithmetic Progression (A.P.). Therefore, are in A.P. The condition for A.P. gives: .

Step 2: Simplify the right side using a common denominator. Combine the fractions on the right side: .

Step 3: Apply trigonometric sum-to-product and product-to-difference formulas. The numerator expands to . The denominator expands to the standard identity . Substituting these back gives: .

Step 4: Cross-multiply and rearrange the terms. Divide both sides by 2 and cross-multiply: . Gather all terms on one side: .

Step 5: Use half-angle identities to solve for . We apply the half-angle identities: and .
Squaring the sine identity gives . Substitute these into the rearranged equation: . Assuming , we divide both sides by to isolate , resulting in .
The question asks for . We know the double-angle identity . Substituting our derived value for into this identity yields . $ $

Concept: Trigonometry - Properties of Triangles (Sine Rule) and Arithmetic Progressions.

Step 1: Determine angle B from the Arithmetic Progression property. Given that angles A, B, and C are in an Arithmetic Progression (A.P.), the middle term is the average of the other two: .
In any triangle, the sum of interior angles is : . Substitute with : . Therefore, angle .

Step 2: Apply the Sine Rule to the given expression. The Sine Rule states that the ratio of a side length to the sine of its opposite angle is constant: . This implies that is proportional to , and is proportional to . Thus, we can rewrite the ratios as and .
Substitute these into our target expression: .

Step 3: Expand the double angles using trigonometric identities. Use the double angle formula for sine, , to expand the expression: .

Step 4: Cancel terms and simplify the expression. In the first term, cancels out from the numerator and denominator. In the second term, cancels out. This leaves: . Factor out the 2: .

Step 5: Use the angle sum formula to evaluate. The expression inside the parentheses exactly matches the angle addition formula for sine: .
So, . From Step 1, we know . Since , .
Substitute this angle: . Using standard trigonometric values, . Evaluate: . $ $

Concept: Trigonometry - Solving Trigonometric Equations for Principal Solutions.

Step 1: Determine the domain restrictions. The given equation contains and , which are undefined when . Therefore, the solutions cannot be odd multiples of (e.g., ).

Step 2: Convert the equation entirely into sine and cosine. Rewrite the original equation in terms of sine and cosine: . Combining the fractions yields .

Step 3: Cross-multiply and form a quadratic-style equation in sine. Multiply both sides by : . To solve this, we must express everything in terms of a single trigonometric function. Use the Pythagorean identity to substitute for : .

Step 4: Rearrange into a standard quadratic equation and factor. Move all terms to one side to form a quadratic equation in terms of : . Factor the quadratic equation: . This gives .

Step 5: Find the principal solutions, respecting domain restrictions. From the factored form, we have two possibilities: or .
However, if , then (from ). From Step 1, we know because it makes the original equation undefined. Thus, we reject .
We are left with . Since sine is positive in the first and second quadrants, the principal solutions (solutions between and ) are and . $ $

Step 1: Use the cosine rule.
Using the cosine rule in triangle ABC: Given , we know that . Substituting the values of , , and , we can solve for the value of .
Step 2: Simplification.
After simplifying, we find that . Therefore, the correct answer is .
Step 3: Conclusion.
The correct answer is (D) 90°.

Step 1: Use the identity for .


Step 2: Substitute the given value.


Step 3: Simplify.


Step 4: Conclusion.

Step 1: Apply the sine rule.
Using the sine rule in :
Step 2: Substitute the given values.

Step 3: Simplify the equation.

Step 4: Find angle .

Step 1: Understanding the exponential growth.
The problem describes exponential growth, which follows the equation: where is the initial number of bacteria, is the rate constant, and is the time.

Step 2: Using the doubling time.
Since the number of bacteria doubles in 4 hours, we have the relation: Solving for , we get , so .

Step 3: Finding the number of bacteria after 12 hours.
Using the equation , we calculate the number of bacteria after 12 hours: Thus, the number of bacteria after 12 hours is , which makes option (C) the correct answer.

Step 1: Understand the condition for the bisector.
For the equation of the conic, the condition for one of the lines being the bisector of the angle between the axes is given by . This can be derived by considering the condition for the angle between two lines and equating it to the angle between the coordinate axes.

Step 2: Conclusion.
Thus, the correct answer is , corresponding to option (A).

Step 1: Use the identity for .
We know that: Substitute into the equation: Thus,
Step 2: Determine the correct sign.
Since lies in the third quadrant, both and are negative, so must also be negative. Therefore,
Step 3: Conclusion.
Thus, the value of is , corresponding to option (A).

Step 1: Simplify the expression.
We can first simplify both parts of the expression separately. Consider the second term . Using the identity: So, We know that , and , so: Step 2: Simplify the first part of the expression.
For the first part, use the identity for simplifying the sum and difference of sine and cosine: which simplifies further using trigonometric identities. Step 3: Conclusion.
After simplifying both terms, the final answer is .

Step 1: Use the tangent subtraction identity.
We know that Step 2: Rewrite the given equation.
Step 3: Use the standard value of inverse tangent.
Step 4: Solve for .
Step 5: Conclusion.
Since , the required value is

Step 1: Expand the cosine term.
Step 2: Expand the sine term.
Step 3: Subtract the expressions.
Step 4: Simplify.
Step 5: Conclusion.
Hence, the simplified expression is .

Step 1: Apply the cosine rule.
For triangle ,

Step 2: Substitute the given values.


Step 3: Rearrange the equation.


Step 4: Conclusion.
Hence, satisfies the equation

Step 1: Express and in terms of and .
We are given two equations: Adding these two equations: Subtracting the two equations: Step 2: Find and .
We know that and . For : For : Step 3: Conclusion.
Thus, the values of and are and , respectively.

Step 1: Simplify the angles.
is equivalent to , and is equivalent to .
Step 2: Calculate the result.
After simplifying, we find that: Thus, .
Step 3: Conclusion.
The correct answer is (A) 1.

Step 1: Use standard trigonometric identity.
We know that
Step 2: Substitute .

Step 3: Final result.
Thus,

Step 1: Analyze the given condition.
The given condition is . By using the sine rule in the triangle and simplifying the equation, we can find that . This shows that sides and are equal.

Step 2: Conclusion.
Thus, the correct answer is , corresponding to option (C).

Step 1: Use trigonometric identities.
We apply the sum identities for sine and cosine: and Step 2: Substitute the values.
Using , , , and , we get: Step 3: Conclusion.
The correct answer is (D) .

Step 1: Use the sum of angles identity.
We are given the equations for and . By using the sum of angles identity for sine and cosine, we find that:
Step 2: Conclusion.
Thus, , corresponding to option (B).

Step 1: Use half-angle identities.
where .

Step 2: Substitute in the given expression.


Step 3: Simplify.
But , hence

Step 4: Conclude the relation among sides.
Thus, are in arithmetic progression.

Step 1: Express and .
We are given . Using the identity , we square both sides: Expanding the left-hand side: Using the identity , we substitute and simplify to find .
Step 2: Solve for .
After finding , we use to compute , yielding . Thus, the correct answer is option (B).

Step 1: Calculating the mean .

Step 2: Calculating .

Step 3: Finding the variance.

Step 4: Finding the standard deviation.

Step 5: Conclusion.
The standard deviation of is .

Step 1: Use right triangle relations.
Since , we have Step 2: Use the identity for sum of inverse tangents.
Step 3: Apply the formula.
Let Step 4: Compute numerator and denominator.
Step 5: Simplify using .
After simplification, Step 6: Conclusion.
Hence, the required value is .

Step 1: Using the condition of continuity at .
For continuity at ,
Step 2: Evaluating the limit.

Step 3: Equating with .

Step 4: Conclusion.
The value of is .

Step 1: Check continuity at .
For continuity at , we check the limit of as . We find that and , so the function is not continuous at .

Step 2: Check differentiability at .
Since the function is not continuous at , it cannot be differentiable there.

Step 3: Conclusion.
Thus, the function is neither continuous nor differentiable at , corresponding to option (C).

Step 1: Use the product-to-sum identity.


Step 2: Apply the identity to each term.


Step 3: Subtract the expressions.


Step 4: Use the identity for difference of cosines.


Step 5: Simplify the expression.


Step 6: Conclusion.

Step 1: Using the condition for coplanarity.
Three vectors are coplanar if and only if their scalar triple product is zero. The scalar triple product of vectors is given by:
Step 2: Substituting the vectors.
Let the vectors be: We compute the cross product and then take the dot product with . After solving, we find that .

Step 3: Conclusion.
Thus, , which makes option (C) the correct answer.

Step 1: Use the Law of Sines.
Given that the angles of are and , we can use the Law of Sines to relate the sides of the triangle. The law states that: where are the angles and are the corresponding opposite sides.
Step 2: Find the ratio of the smallest and greatest sides.
By applying the Law of Sines, we find that the ratio of the smallest side to the greatest side is . Thus, the correct answer is option (B).

Step 1: Find basic trigonometric ratios.
Since is in the fourth quadrant,

Step 2: Substitute in the expression.


Step 3: Simplify.


Step 4: Conclusion.

Step 1: Use the given equation.
We are given the equation: Simplifying the right-hand side: Thus, the equation becomes:
Step 2: Use the law of cosines.
Now, applying the law of cosines and the known values for , , and , we find the value of . Using , we substitute back into the equation:
Step 3: Conclusion.
Thus, the value of is 12, which corresponds to option (D).

Step 1: Understanding the number of relations.
The number of relations from set to set is given by the formula: where is the number of elements in set and is the number of elements in set . In this case, and . Thus, the total number of relations is:
Step 2: Conclusion.
Thus, the total number of relations is 64, which makes option (A) the correct answer.

Step 1: Use cosine rule relations.
In triangle , Step 2: Compute .
Hence, Step 3: Add .
Using , we get Step 4: Simplify.
Adding both terms,

Step 1: Use substitution.
Let , so that . The integral becomes:
Step 2: Break the integral.
We can separate the terms in the numerator:
Step 3: Perform the integration.
The first integral is , and the second integral is . Substituting , we get the final answer:
Step 4: Conclusion.
Thus, the correct answer is , corresponding to option (B).

Step 1: Find the derivative of the curve.
To find the equation of the normal line, we first compute the derivative of to obtain the slope of the tangent at the point . The derivative is . At , the slope of the tangent is 0.

Step 2: Find the slope of the normal.
The slope of the normal line is the negative reciprocal of the tangent slope. Since the tangent slope is 0, the normal slope is undefined, which implies that the normal line is a vertical line. Thus, the equation of the normal is .

Step 3: Conclusion.
Thus, the equation of the normal is , corresponding to option (C).

Step 1: Writing the parametric form of the line.
Let Then
Step 2: Substituting in the plane equation.

Step 3: Finding the point of intersection.

Step 4: Conclusion.
The point of intersection is .

Step 1: Evaluate .
We know that .
Step 2: Evaluate .
We know that .
Step 3: Evaluate .
We know that .
Step 4: Add the results.
Step 5: Conclusion.
Hence, the value is .

Step 1: Simplify each trigonometric function.
- For , - For , - For , - For , Step 2: Identify the matching pairs.
We find that Thus, only (C) and (D) have the same value.

Step 1: Plot the constraints and find the feasible region.
The constraints are: - - - - The feasible region is the region where all the inequalities are satisfied.
Step 2: Calculate the corner points.
By solving the system of equations formed by the constraint lines, we find the corner points of the feasible region.
Step 3: Evaluate the objective function at the corner points.
Substitute the corner points into to find the optimal value of . The point gives the maximum value of .
Step 4: Conclusion.
The optimal solution is .

Step 1: Apply the chain rule.
We are asked to evaluate the integral . First, we notice that the integrand is the product of an exponential function and a rational function. Let's simplify and integrate each part.
Step 2: Simplify the expression.
We can rewrite the integrand as: This can be seen as a standard form of an integral involving and the quotient .
Step 3: Solve the integral.
The integral simplifies to:
Step 4: Conclusion.
The correct answer is .

Step 1: Use the tangent addition formula.

Step 2: Substitute the given values.

Step 3: Find .

Step 1: Use the cosine rule.
Using cosine rule in :
Step 2: Simplify numerator and denominator.

Step 3: Take ratio.

Step 1: Use trigonometric identities.
We are given the equation: Using trigonometric identities for the sine and cosine of sums and differences, we simplify both sides of the equation.
Step 2: Simplify and solve.
After simplifying, we find that . Thus, the correct answer is option (D).

Step 1: Understanding the equation.
We are given . Using the identity , we equate it to . This leads to multiple possible angles for , and we derive the general solution.

Step 2: Analyzing the options.
Option (D) correctly matches the derived formula for , which is . This formula reflects the periodic nature of trigonometric functions.

Step 3: Conclusion.
Thus, the correct answer is option (D).

Step 1: Direction ratios of the given lines.
Let the direction ratios of the given lines be and . The direction ratios of the line perpendicular to both of these lines can be found by taking the cross product of and .

Step 2: Computing the cross product.
The cross product of and is: After calculating the determinant, we find the direction ratios of the perpendicular line as .

Step 3: Conclusion.
Thus, the direction cosines of the perpendicular line are , which makes option (B) the correct answer.

Step 1: Understanding the geometry of the quadrilateral.
In any quadrilateral, if and are the mid-points of sides and respectively, the segment is known as the mid-line. The mid-line theorem states that the length of is half the length of the sum of the opposite sides, and .

Step 2: Applying the mid-line theorem.
From the mid-line theorem, we know that: This gives us the equation , which means .

Step 3: Conclusion.
Thus, the correct value of is 2, which makes option (B) the correct answer.

Step 1: Approximation using small angle.
Since is close to 1, we can use the approximation for small values of , where . In this case, the value of .
Step 2: Conclusion.
Thus, the approximate value is , corresponding to option (A).

Step 1: Use the given condition.
The given condition is , which is a known property of right-angled triangles. This condition holds specifically for a right-angled triangle.
Step 2: Conclusion.
Thus, is a right-angled triangle, corresponding to option (C).

Step 1: Use the given equation.
We are given the equation: By solving the equation, we find that .
Step 2: Conclusion.
Thus, the value of is , corresponding to option (A).

Step 1: Understand the expected value formula.
The expected value of a discrete random variable is given by the sum of all possible outcomes weighted by their probabilities. For a die, the possible outcomes are 1, 2, 3, 4, 5, and 6, and the probability of each outcome is .
Step 2: Calculate the expected value.
The expectation of the number on the die is: Thus, the expected value is , which corresponds to option (B) .
Step 3: Conclusion.
Thus, the correct answer is , corresponding to option (B).

Step 1: Use the inverse sine function.
For , the principal value lies in the interval . We know that . Therefore, the principal value is .
Step 2: Conclusion.
The correct answer is (D) .