Trigonometric Equations

Given,

Step 1: Use trigonometric identities.
We know Substitute into the given equation:
Step 2: Simplify the equation.
Multiply both sides by 2:
Step 3: Rearrange terms.
or
Step 4: Write in standard form.
where Hence,
Step 5: Solve for .
This gives
Step 6: Check the given condition.
The condition removes extraneous solutions, leaving

Step 1: Express in terms of .
We know that
Step 2: Substitute in the given equation.

Step 3: Simplify the quadratic equation.

Step 4: Solve for .

Step 5: Choose the permissible value.
Since ,

Step 1: Solve the given trigonometric equation.
The given equation is: We use the tangent addition formula to simplify: So, we first express as:
Step 2: Analyze the solutions.
By solving this equation, we get:
Step 3: Conclusion.
Thus, the general solution is , corresponding to option (B).

Step 1: Express cos 248° and sin 212° in terms of known angles:
So, the expression becomes:

Step 2: Express cos 68° and sin 32° in terms of known angles:
Using the identity , we simplify:

Step 3: Simplify the expression using trigonometric identities:
We use the identity for the difference of sines: Applying this identity:

Step 4: Use a calculator to find the numeric values:
Using a calculator, we find: Thus: The expression evaluates to approximately 0.1553.

Final Answer:
Therefore, the final answer is approximately:

Given Equation:
We are given the equation: and we substitute into the equation. We will simplify it step by step.

Step 1: Substitute :
Substituting into the equation, we get:

Step 2: Simplify the expression:
Now, simplifying further:

Step 3: Simplify further:
We continue to simplify this expression:

Step 4: Combine the terms:
Simplifying further, we have:

Step 5: Simplify the final expression:
The equation simplifies to:

Step 6: Use a trigonometric identity:
This can be rewritten as:

Step 7: Simplify further:
Simplifying the above equation:

Step 8: Solve for :
From this equation, we can deduce that: Thus:

Step 9: Find the relation with :
Since , we can conclude that:

We are tasked with evaluating the integral:

Step 1: Rewrite the integral: The given integral is: This expression can be rewritten as: Since , we can simplify this to:

Step 2: Simplify each integral: Now, simplify the individual integrals: These integrals can be recognized as standard trigonometric integrals. The first integral is the integral of , and the second is the integral of :

Step 3: Integrate: The integral of is , and the integral of is . Thus:

Final Answer: Therefore, the integral is:

The equation is a product of two factors. For the equation to be zero, either or .
- leads to , where is an integer.
- simplifies to , which gives solutions .
Thus, the principal solutions are and .

The principal solution refers to a solution that satisfies a given equation or system of equations. In this case, the given expression consists of two factors, and the correct identification of the principal solution can be derived by analyzing the components separately.
- Option (1) is incorrect because it is a simple sinusoidal function. The expression is part of the equation but may not be the principal solution.
- Option (2) is correct as it contains a cosine function, contributing to the solution structure.

Concept: A useful trigonometric identity is: This implies Thus, we solve the equation by considering both possible cases.
Step 1: {Apply the identity .} Given This gives two cases: or
Step 2: {Solve the first case.}
Step 3: {Solve the second case.}
Step 4: {Combine the solutions.} Thus, the general solution is where .