Trigonometric Equations

The correct answer is 4
The given equation is sin x = cos² x
This can be stated as
sin²x + sinx-1 = 0
Use the quadratic formula

Only 4 roots
Consider the following graph :

Therefore , the number of solutions for this equation is 4 solutions.

The given equation is:

Simplify using trigonometric identities:

Solve for , leading to:

There are positive and negative values:

Final Answer:

The correct answer is option (C): 3

Given,

as

The number of solutions is 3.

The correct answer is option (C): 3

Given,

as

The number of solutions is 3.

The given equation is:

Rewriting the left-hand side using the sum-to-product identities:

From the identity , we have:

Substituting back:

Simplify further:

This shows:

Given , substitute , and .

Now compute:

Substitute the values:

Simplify:

Convert to a common denominator:

Simplify further:

Given:


Simplify :


Re(z) = 0:


Solving for :


Set of Angles:


Sum:

To determine the domain of , the following conditions must be satisfied:

1. Condition for the logarithmic term: The argument of the logarithmic function must be positive:

Analyze the sign changes of the numerator and denominator:

The critical points divide the real line into intervals:

Test the sign of in each interval to determine positivity:

2. Condition for the inverse cosine term:

The argument of must lie in the interval :

Solve the inequality:

Analyze the critical points and test the valid ranges.

3. Combine the results:

The final domain of is , where:

4. Calculate :

Substitute the values:

Calculate:

Final Answer:

λ = cos²2x - 2sin⁴x - 2cosx

Convert all in to cos x.

λ = (2cos²x - 1)² - 2(1 - cos²x)² - 2cosx

= 4cos⁴x - 4cos²x + 1 - 2(1 - 2cos²x + cos⁴x) - 2cosx

= 2cos⁴x - 2cos²x + 1 - 2

= 2cos⁴x - 2cos²x - 1

= 2 [cos⁴x - cos²x - 1/2]

= 2 [(cos²x - 1/2)² - 3/4]

λmax = 2 [1 - 3/4] = 2 × (2/4) = -1 (max value)

λmin = 2 [0 - 3/4] = -3/2 (minimum value)

So, Range = [-3/2, -1]

Given:

Let:

Rewriting :

Step 1: Evaluate :

Substitute into :

Simplify:

Expand numerator and denominator:

Numerator:

Denominator:

Thus:

Step 2: Evaluate :

Step 3: Evaluate :

Step 4: Apply AM-GM Inequality: Using AM-GM:

Final Answer:

We are tasked with simplifying the expression using the given formula:

Step 1: Identify the Parameters

The expression becomes:

Using the formula:

Step 2: Simplify Using the Formula

Substitute :

Step 3: Evaluate the Terms

Using the trigonometric identity , we know:

Substitute this back into the equation:

The terms cancel out, leaving:

Final Answer:

The simplified result is:

The correct answer is option (D) : More than 2

We are given the function and we need to solve for based on the given conditions.
Step 1: Simplifying the function .
We start by simplifying the expression for : Using the trigonometric identities: we get: Simplify further: Thus, the simplified expression for is:

Step 2: Finding the derivative of . Now, we differentiate with respect to : This simplifies to:

Step 3: Solving for when . We are given that . Set the equation for equal to : Solve this equation for . After solving, we find that the set of solutions for is:

Step 4: Calculating . Now, we calculate using the given condition : Thus,

Step 5: Calculating .
Finally, we calculate . Using , we substitute this into the simplified expression for : Simplifying, we get: Thus, the value of is , and the correct answer is option (3).

The given problem involves solving the trigonometric equation for a specific range of , where . We need to find the value of .

We can express the equation using the form , where is the resultant amplitude and is the phase shift.

First, we find :

Now, express and in terms of :

Comparing the coefficients, we have:

and

Using these, we can find :

Hence, the equation becomes:

Therefore:

Thus, the angle is such that:

By the identity for calculating tangent from sine and cosine:

From the identity:

,

Using these identities, we can find the :

Simplifying, we find that the value of matches , which is option (3).

Therefore, the correct answer is .

Given: Find .

Concept Used:

Tangent addition formula: Also, verify if or etc.

Step-by-Step Solution:

Step 1: Write down and .

Step 2: Compute using the formula.

Step 3: Simplify numerator and denominator separately.

Numerator: Denominator:

Step 4: Divide numerator by denominator.

Cancel (since ):

Step 5: Compare with .

Given: Factor : So: Thus:

Step 6: Compare and .

So: Since , we have: But the problem likely expects or . Let's check: ⇒ .

Step 7: Final answer.

But the options might just say .

Given the symmetry, the correct relation is: So .

Therefore, .

We are asked to find the total number of solutions for the trigonometric equation within the interval .

Concept Used:

The given equation is a quadratic equation in the form of , where . We will solve this quadratic equation for using the quadratic formula:

After finding the possible values for , we will determine the corresponding values of in the interval .
The number of solutions will be the count of all such distinct values of .

Step-by-Step Solution:

Let . The given equation can be rewritten as a quadratic equation in :

Here, the coefficients are , , and .
We apply the quadratic formula to solve for .

First, we calculate the discriminant, :

To simplify , we try to express as a perfect square of the form . We can write .
We look for two numbers whose sum is 10 and product is 24.
These numbers are 6 and 4. So,

Therefore, .

Now, we substitute this back into the quadratic formula:

This gives two possible values for :

Case 1: Using the '+' sign.

Case 2: Using the '-' sign.

Now we find the number of solutions for each case in the interval .

For :

In the interval , is positive in the first and fourth quadrants.
The solutions are and .
In the interval , the solutions are and .
Thus, there are 4 solutions for this case: .

For :

In the interval , is negative in the second and third quadrants.
The solutions are and .
In the interval , the solutions are and .
Thus, there are 4 solutions for this case: .

Final Computation & Result:

The total number of solutions is the sum of the number of solutions from both cases.
Since the values and are distinct, there are no overlapping solutions.

Number of solutions from Case 1 = 4.

Number of solutions from Case 2 = 4.

Total number of solutions = 4 + 4 = 8.

Hence, the total number of solutions of the given equation in the interval is 8.

To solve the problem of finding the number of solutions for the equation in the interval , we start with the given equation:

Let's approach this problem step-by-step:

First, simplify the right side of the equation using the trigonometric identity for :

Substitute this in the equation:

Rearrange the terms to formulate a new equation:

This simplifies our target equation.

For simplicity in finding the solutions, consider analyzing it graphically or by identifying particular solutions through trial in the given interval. We need to check for how many values of in the interval the two sides of the equation are equal.

We can approximate the solution count through visualizing the periodic behavior of trigonometric functions involved. Being complex in analytical derivation, solving numerically or graphically gives a straightforward estimate. The function will iterate over several solutions due to the periodic nature of cosine in such transformations.

An evaluation reveals that there are 7 solutions to the equation in the given interval.

Thus, the correct answer is 7.

To solve the given problem, we need to determine the set of angle values that satisfy the condition: and then evaluate .

First, let's simplify the expression:

The given complex expression is .

To find its real part, we multiply by the conjugate of the denominator:

Calculating the denominator:

Now, calculating the numerator:

This simplifies to:

Thus, the real part of the expression is:

Setting the real part to zero, we have:

This implies:

Solving for , we get:

Therefore, , which implies or or or .

Now, we compute using these values:

Since , we substitute:

Since the problem is symmetric and involves calculations, the sum is equivalent to based on the symmetry and periodic nature of and functions over the specified interval.

Thus, .

Step 1: Understand the Parametric Equations

The points are given by:

Step 2: Use Trigonometric Identities

Let:

Thus, . Using the tangent of a difference formula:

Given , we have:

Step 3: Simplify the Equation

Simplify numerator and denominator:

Multiply both sides by :

Multiply by :

Rearrange terms:

Step 4: Compare with Given Curve

The given curve is:

Comparing coefficients:

Sum them up:

Step 6: Match with Options

The correct answer corresponds to option (4).