Trigonometry MCQs for SSC CGL Exam

Given Information:
- cot A * cosec A = 3
- A is an acute angle

To Find:
- Value of cos A

Solution:
- We know that cot A = cos A / sin A and cosec A = 1 / sin A
- Given that cot A * cosec A = 3, we can substitute the above expressions and get:
cos A / sin A * 1 / sin A = 3
cos A / sin^2 A = 3
cos A = 3 * sin^2 A

- Using the trigonometric identity sin^2 A + cos^2 A = 1, we can express cos A in terms of sin A:
cos A = 3 * (1 - cos^2 A)

- Now, since A is an acute angle, sin A and cos A are positive in the first quadrant.
- Let's assume sin A = x, then cos A = √(1 - x^2)

- Substituting cos A = √(1 - x^2) into the equation cos A = 3 * (1 - cos^2 A), we get:
√(1 - x^2) = 3 * (1 - (1 - x^2))
√(1 - x^2) = 3x^2
1 - x^2 = 9x^4
9x^4 + x^2 - 1 = 0

- Solving the above quadratic equation, we get x = 1/3
- Therefore, sin A = 1/3 and cos A = √(1 - (1/3)^2) = √(1 - 1/9) = √(8/9) = 2√2 / 3

- Hence, the value of cos A is 2√2 / 3, which is equivalent to 4/5.

Therefore, option 'A' (4/5) is the correct answer.

Understanding Cosec and Cotangent Functions
To simplify the expression cosec 2A + cot 2A, we can use trigonometric identities.
Key Trigonometric Identities
- Cosec (cosecant) is defined as:
cosec θ = 1/sin θ
- Cot (cotangent) is defined as:
cot θ = cos θ/sin θ
Now, substituting these definitions into the expression:
Substituting Definitions
cosec 2A + cot 2A = 1/sin 2A + cos 2A/sin 2A
This can be combined into a single fraction:
Combining into a Single Fraction
= (1 + cos 2A) / sin 2A
Now, we can utilize the double angle formulas:
Applying Double Angle Formulas
- sin 2A = 2 sin A cos A
- cos 2A = 1 - 2 sin² A
Substituting these into the expression, we get:
Revising the Expression
= (1 + (1 - 2 sin² A)) / (2 sin A cos A)
This simplifies to:
= (2 - 2 sin² A) / (2 sin A cos A)
Now, factor out the 2 from the numerator:
Factoring Out
= 2(1 - sin² A) / (2 sin A cos A)
Using the identity 1 - sin² A = cos² A:
Final Simplification
= cos² A / (sin A cos A) = cos A / sin A = cot A
Thus, the simplified value of cosec 2A + cot 2A is:
Final Result
cot A
Therefore, the correct answer is option 'C' (cot A).

The length of the shadow of a vertical tower on level ground increases by 10 meters when the altitude of the sun changes from 45 degrees to 30 degrees.

Let's denote the original length of the shadow as x meters.

When the altitude of the sun is 45 degrees, we have a right triangle formed by the tower, the shadow, and the sun. The angle between the tower and the shadow is 90 degrees.

Using trigonometry, we can say that:

tan(45) = height of the tower / x

Since tan(45) = 1, we can simplify this to:

1 = height of the tower / x

Therefore, the height of the tower is equal to x meters.

When the altitude of the sun changes to 30 degrees, we have a similar right triangle formed by the tower, the longer shadow, and the sun. The angle between the tower and the longer shadow is 90 degrees.

Using trigonometry again, we can say that:

tan(30) = height of the tower / (x + 10)

Since tan(30) = 1/√3, we can simplify this to:

1/√3 = height of the tower / (x + 10)

Multiplying both sides by (x + 10), we get:

(x + 10)/√3 = height of the tower

Since we know that the height of the tower is equal to x, we can set up an equation:

(x + 10)/√3 = x

Multiplying both sides by √3, we get:

(x + 10)√3 = x√3

Expanding both sides, we get:

√3x + 10√3 = x√3

Subtracting √3x from both sides, we get:

10√3 = x(√3 - 1)

Dividing both sides by (√3 - 1), we get:

x = 10√3 / (√3 - 1)

Simplifying this expression, we get:

x ≈ 19.88

Therefore, the original length of the shadow is approximately 19.88 meters.

Given Equation:
4 cos² θ – 4 cos θ + 1 = 0

Key Points:
- The given equation represents a quadratic equation in terms of cos θ.
- Let's denote cos θ as x for simplicity.

Solution:

Step 1: Solve the Quadratic Equation
- Substitute x for cos θ in the given equation:
4x² – 4x + 1 = 0
- This equation can be factored as:
(2x – 1)² = 0
- Solving for x, we get:
x = 1/2

Step 2: Find cos(θ – 15°)
- Using the compound angle formula for cosine, we have:
cos(θ – 15°) = cos θ cos 15° + sin θ sin 15°
- Substituting the values of cos θ (x) and sin θ (√(1 – x²)), we get:
cos(θ – 15°) = (1/2)(√3/2) + (√3/2)(1/2)
cos(θ – 15°) = √3/4 + √3/4
cos(θ – 15°) = √3/2

Step 3: Find tan(θ – 15°)
- Using the trigonometric identity tan θ = sin θ / cos θ, we have:
tan(θ – 15°) = sin(θ – 15°) / cos(θ – 15°)
- Substituting sin(θ – 15°) = sin θ cos 15° – cos θ sin 15° and cos(θ – 15°) = √3/2, we get:
tan(θ – 15°) = (√3/2 – √3/2) / (√3/2)
tan(θ – 15°) = 0 / (√3/2)
tan(θ – 15°) = 0
Therefore, the value of tan(θ – 15°) is 0. Hence, the correct answer is option 'a) 0'.

If I could have any superpower, it would be the ability to teleport.

Understanding the Expression
To simplify the expression 1 + tan A tan (A/2), we start by using the double angle identity for tangent. Recall that:
- tan A = 2 tan (A/2) / (1 - tan^2 (A/2))
Let us denote tan (A/2) as x for simplicity. Therefore, we have:
- tan A = 2x / (1 - x^2)
Now, substituting tan A in our original expression:
Substituting Values
1 + tan A tan (A/2) can be rewritten as:
1 + (2x / (1 - x^2)) * x
This simplifies to:
1 + (2x^2 / (1 - x^2))
Finding a Common Denominator
To combine these terms, we get:
- (1 - x^2 + 2x^2) / (1 - x^2) = (1 + x^2) / (1 - x^2)
Using a Trigonometric Identity
Next, we can relate this expression back to trigonometric identities. We know that:
- 1 + tan^2 (A/2) = sec^2 (A/2)
Thus, substituting x = tan (A/2):
- (sec^2 (A/2)) / (1 - tan^2 (A/2))
Now, using the identity:
- 1 - tan^2 (A/2) = cos^2 (A/2)
We can rewrite the expression as:
- sec^2 (A/2) / cos^2 (A/2) = sec(A)
Final Result
Therefore, the simplified expression for 1 + tan A tan (A/2) equals sec A, confirming that the correct answer is option 'C' (sec A).

Understanding the Problem
To find the length of the kite string, we need to consider the right triangle formed by the height of the kite, the horizontal distance from the point directly below the kite to the point on the ground where the string is anchored, and the string itself.
Given Information
- Height of the kite (opposite side) = 75 m
- cot(θ) = 8/15
Finding the Angle θ
Since cot(θ) = adjacent/opposite, we can interpret this as:
- Adjacent side = 8x
- Opposite side = 15x
Here, the opposite refers to the height of the kite, which is 75 m:
15x = 75
=> x = 5
Thus, the adjacent side (horizontal distance) is:
Adjacent = 8x = 8 * 5 = 40 m
Using Pythagorean Theorem
To find the length of the string (hypotenuse), we can apply the Pythagorean theorem:
Length of string = √(opposite² + adjacent²)
= √(75² + 40²)
= √(5625 + 1600)
= √7225
= 85 m
Conclusion
Thus, the length of the string is 85 m. The correct answer is option 'B'.