Inverse Trigonometric Functions

Step 1: Concept
Formula: .
Step 2: Analysis
Given .
Step 3: Calculation
The tangent of is undefined (approaches infinity), which means the denominator of the fraction must be zero.
.
or .
Step 4: Conclusion
Hence, the correct relation is .
Final Answer:(B)

To find the value of given by , we'll follow these steps:

Let . Then .

This implies .

Using the Pythagorean identity, the hypotenuse is .

Thus, .

Let . Then , which implies .

The hypotenuse is .

Then, .

Cross-multiply: .

Divide both sides by 4: .

Square both sides: .

.

Subtract 1: .

.

Take square root: .

The value of is , which matches the provided correct answer.

We begin by recognizing standard angle values:

So:

To solve the problem of determining the value of , we will evaluate each term individually and then add them together.

1. First, calculate :
In a right triangle where the opposite side is and the hypotenuse is , the adjacent side, calculated by the Pythagorean theorem, is (because ). Thus, , where .

2. Next, calculate :
In a right triangle where the adjacent side is and the hypotenuse is , the opposite side is (since ). Thus, , where .

3. Lastly, calculate :
This represents an angle where the opposite side is and the adjacent side is , so the tangent is .

Now sum the angles. Using the identities:

We find:

The angle such that corresponds to the angle whose tangent equals the division of opposite by adjacent sides in a triangle, completing the gap to reach .

So, the sum of the three angles provides:

Thus, the correct answer is .

To find the value of , follow these steps:

1. Start with the expression . We need to determine the angle such that .

2. The angle for which is .

3. Now, substitute into the cosine function: .

4. Calculate: .

5. We need to find . The angle such that is .

6. Therefore, the value of is .

Let . According to the tangent addition formula: Given the values and , we have: Simplifying the expression, we have: Simplifying the numerator: Simplifying the denominator: Thus, substituting back, we have: Therefore, .

To solve the equation , we can use the tangent addition formulas. First, let's denote and . Then the equation becomes .

Using the tangent double angle identity: , where implies . Hence, .

Now .

Use the tangent addition formula: . Applying it: .

Calculate denominator: .

Therefore, .

So, , which confirms that the value of is .

To determine the value of , we begin by recognizing that if , then . We need to find given .

Recall the identity for tangent in terms of sine and cosine:

Since , we need . Using the Pythagorean identity:

Substitute :

Thus,

Taking the square root, we get:

Now substitute and into the tangent formula:

Therefore, the value of is , which matches the given option .

To solve the expression , we follow these steps:

The principal value of is determined by finding the angle such that:

Recall that .

Since satisfies the condition and corresponds to the sine function result , it follows that:

The principal value of is indeed .

Therefore, the correct answer is:

The given equation is .

Since , we have:

Simplify the second term:

The equation becomes:

The property of tangent gives us , so this equation implies:

This term is undefined, indicating the denominator must be zero:

Cross-multiply to eliminate fractions:

Combine like terms:

To confirm both roots, check when the condition fails. Solving :

Cross-multiplying gives:

Simplifies to:

Thus, the correct answer is: There is one positive and one negative real value of satisfying the above equation.

Step 1: Understanding the Concept:
This problem involves finding the value of a trigonometric function of an inverse trigonometric function. A common method is to use a right-angled triangle to represent the inverse trigonometric function and then find the required trigonometric ratio.
Step 2: Key Formula or Approach:
Let . This implies . We need to find .
We can visualize this relationship using a right-angled triangle.
Step 3: Detailed Explanation:
Let . Then .
We can write .
In a right-angled triangle, .
So, we can let the side opposite to angle be and the adjacent side be 1.
Using the Pythagorean theorem, we can find the hypotenuse:
Now, we need to find , which is .
The formula for is .
Step 4: Final Answer:
Therefore, .

Step 1: Understanding the Concept:
This problem involves finding the value of a trigonometric function of an inverse trigonometric function. A common method is to use a right-angled triangle to represent the inverse trigonometric function and then find the required trigonometric ratio.
Step 2: Key Formula or Approach:
Let . This implies . We need to find .
We can visualize this relationship using a right-angled triangle.
Step 3: Detailed Explanation:
Let . Then .
We can write .
In a right-angled triangle, .
So, we can let the side opposite to angle be and the adjacent side be 1.
Using the Pythagorean theorem, we can find the hypotenuse:
Now, we need to find , which is .
The formula for is .
Step 4: Final Answer:
Therefore, .