Some Applications Of Trigonometry

From the above figure:

The answer is 1.

Step 1: Use the addition formula for arctangents.
We use the formula for the sum of two arctangents: So the equation becomes: Step 2: Solve for .
We have: This implies: Step 3: Solve the resulting equation.
Now, solve for . This will yield only one solution that satisfies the given condition. Step 4: Conclusion.
Thus, the number of values of that satisfy the equation is 1. Final Answer:

Step 1: Simplify the expression.
The given equation involves trigonometric identities. First, simplify the numerator and denominator separately using standard trigonometric identities. You can use the identity and others as necessary. Step 2: Applying the values.
After applying the trigonometric simplifications and solving for and , we find that: Step 3: Conclusion.
Thus, . Final Answer:

Step 1: Write the given expression.
We are given the expression: Step 2: Simplify the expression.
The second term can be written as: Thus, the expression becomes: Step 3: Combine the terms.
Rewriting the expression in a simplified form: Step 4: Substitute the values and calculate.
Using standard trigonometric values for angles, we get the result as 4. Hence, the correct answer is (3) 4.

Step 1: Rewrite the equation using trigonometric identities.
We are given the equation: Recall that: So, the equation becomes: Now multiply both sides by (provided ):
Step 2: Solve the equation.
We need to solve the equation: Using the identity for the tangent of multiple angles, we can solve for the general solutions of within the interval .
Step 3: Count the number of solutions.
The equation has 8 distinct solutions in the interval .
Step 4: Conclusion.
Thus, the number of solutions is 8. The correct answer is (4).

Step 1: Use the identity for inverse trigonometric functions.
We are given the expression , and we need to minimize it in the given interval for . We use the identity: which holds for all . This means that:
Step 2: Simplify the expression.
Expanding the terms: Now, to minimize this expression, we need to consider the values of in the given interval.
Step 3: Calculate the value at the boundary.
We are given the interval .
- For , we calculate .
- For , we calculate .
Thus, the minimum value of the expression occurs when , yielding the value of the expression as: Therefore, the minimum value of is 9.