Trigonometric Ratios

Let’s consider the expression: Multiply both terms by and respectively: We know that: So the expression becomes:

The correct answer is option (A):

Let (since these are angles in a triangle).
Then we know: Now using the identity: We get: Substitute into the expression: Each term becomes 1:

The correct option is (D):

Let the height of the tower , and the horizontal distance .
From triangle ,

Now consider point on such that . So, point is at distance from the base .
In triangle ,

Substitute into the above equation:

Hence, .

The correct option is (C):

We use trigonometric identities:


Numerator: Denominator: So, the expression becomes:

The correct option is (B):

Explanation:
Given: and is acute. This implies that , since . Now, using the double angle identity for sine: Substituting and :

Hence, the value of is .

Correct answer: 1

Explanation:
We know that: So:

Hence, .

Correct answer:

Explanation:
Given: We can use the identity: Substituting the given value of : Now, use the identity for tangent:

Hence, the value of is .

We are given that . We know that the maximum value of the sine function is 1, i.e., for any angle . Therefore, for the sum to equal 3, each of the terms must equal 1. This means: $ \sin \alpha = 1 \alpha = \frac{\pi}{2} + 2n\pi n \sin \beta = 1 \beta = \frac{\pi}{2} + 2m\pi m \sin \gamma = 1 \gamma = \frac{\pi}{2} + 2p\pi p \cos \alpha + \cos \beta + \cos \gamma = 0 + 0 + 0 = 0$.

Step 1: Use identity:
Step 2: Substitute values. So, Now,

Step 1: Given . So, in a right triangle: Using Pythagoras Theorem, hypotenuse =
Step 2: Find

Step 1: Analyze Option (1)
Recall the identity: So, Option (1) is incorrect because it falsely states that:
Step 2: Analyze Option (2)
From the unit circle or sine graph: Hence, Option (2) is true.
Step 3: Analyze Option (3)
For , the value of actually decreases: So, Option (3) is false.
Step 4: Analyze Option (4)
This is not true for all values of , so Option (4) is false.