Welcome to Part 1 of our new 8-part series on Chapter 8, Introduction to Trigonometry. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master trigonometric ratios, identities, and specific angle values.
Recommended Books for Deep Practice
Top 25 MCQs - Introduction to Trigonometry
Question 1: Value of Tan A
Given that sin A = 3/5, the value of tan A is:
Let P = 3k, H = 5k.
By Pythagoras theorem, Base (B) = √(H² - P²) = √((5k)² - (3k)²) = √(25k² - 9k²) = √16k² = 4k.
tan A = Perpendicular / Base = 3k / 4k = 3/4.
Question 2: Identity Application
The value of (1 + tan² θ)(1 - sin θ)(1 + sin θ) is:
(1 - sin θ)(1 + sin θ) = 1 - sin² θ = cos² θ.
So, sec² θ × cos² θ = (1/cos² θ) × cos² θ = 1.
Question 3: Angle Value
If tan A = 1 and sin B = 1/√2, then the value of cos(A+B) is:
sin B = 1/√2 => B = 45°.
So, A + B = 45° + 45° = 90°.
cos(A+B) = cos(90°) = 0.
Question 4: Sec A + Tan A
The value of (sec A + tan A)(1 - sin A) is equal to:
Expression = [(1 + sin A)/cos A] × (1 - sin A)
= (1 - sin² A) / cos A
= cos² A / cos A = cos A.
Question 5: Reciprocal Ratios
If 5 tan θ = 4, then the value of (5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ) is:
Divide numerator and denominator by cos θ:
Num: 5(sin θ/cos θ) - 3 = 5 tan θ - 3
Den: 5(sin θ/cos θ) + 2 = 5 tan θ + 2
Substitute 5 tan θ = 4:
(4 - 3) / (4 + 2) = 1 / 6.
Question 6: Identity 9sec²A
9 sec² A - 9 tan² A is equal to:
Using identity sec² A - tan² A = 1.
9(1) = 9.
Question 7: Angle Evaluation
The value of sin 60° cos 30° + sin 30° cos 60° is:
= 3/4 + 1/4 = 4/4 = 1.
Question 8: Sec A Value
If cos A = 4/5, then the value of tan A is:
P = √(H² - B²) = √(25 - 16) = √9 = 3.
tan A = P/B = 3/4.
Question 9: Trigonometric Table
The value of (sin 30° + cos 30°) - (sin 60° + cos 60°) is:
sin 60° = √3/2, cos 60° = 1/2.
(1/2 + √3/2) - (√3/2 + 1/2) = 0.
Question 10: Max Value
The maximum value of 1/sec θ, where 0° ≤ θ ≤ 90°, is:
The maximum value of cos θ is 1 (at θ = 0°).
Question 11: Complementary Angles
If sin A = 1/2 and cos B = 1/2, then the value of (A + B) is:
cos B = 1/2 => B = 60°.
A + B = 30° + 60° = 90°.
Question 12: Identity Simplification
(1 + tan² A) / (1 + cot² A) is equal to:
Question 13: Sin A = Cos A
If sin θ = cos θ, then the value of 2 tan θ + cos² θ is:
2 tan 45° + cos² 45° = 2(1) + (1/√2)² = 2 + 1/2 = 2.5.
Question 14: Triangle Relation
In ΔABC, right-angled at B, if tan A = √3, then the value of cos A cos C - sin A sin C is:
Since B = 90°, C = 180 - (90+60) = 30°.
cos 60° cos 30° - sin 60° sin 30°
= (1/2)(√3/2) - (√3/2)(1/2) = 0.
(Alternatively, this is the formula for cos(A+C) = cos(90) = 0).
Question 15: Square Identity
If x = a cos θ and y = b sin θ, then b²x² + a²y² is equal to:
= b²a²cos²θ + a²b²sin²θ
= a²b² (cos²θ + sin²θ) = a²b² (1) = a²b².
Question 16: Reciprocal Sum
If tan θ + cot θ = 5, then the value of tan² θ + cot² θ is:
tan²θ + cot²θ + 2(tanθ)(cotθ) = 25
tan²θ + cot²θ + 2(1) = 25
tan²θ + cot²θ = 25 - 2 = 23.
Question 17: Value of expression
The value of (sin 45° + cos 45°) is:
Question 18: Sin²A Identity
If sin A + sin² A = 1, then the value of cos² A + cos⁴ A is:
Substitute sin A for cos² A in the expression:
(sin A) + (sin A)² = 1.
Question 19: Tan Identity
If tan A = 3/4, then cos² A - sin² A is:
cos A = 4/5, sin A = 3/5.
(4/5)² - (3/5)² = 16/25 - 9/25 = 7/25.
Question 20: Complementary
The value of tan 1° tan 2° tan 3° ... tan 89° is:
All pairs cancel out to 1, leaving tan 45° = 1.
1 × 1 × ... × 1 = 1.
Question 21: Cosec Calculation
If √3 tan θ = 3 sin θ, then value of sin² θ - cos² θ is:
sin² θ = 1 - cos² θ = 1 - 1/3 = 2/3.
sin² θ - cos² θ = 2/3 - 1/3 = 1/3.
Question 22: Max Value Sin
The maximum value of sin θ is:
Question 23: Triangle Identity
In ΔABC right angled at B, sin A = 7/25, then cos C is:
cos C = cos(90 - A) = sin A.
Since sin A = 7/25, cos C = 7/25.
Question 24: Sec Value
If sec θ = 25/7, then sin θ is:
P = √(25² - 7²) = √(625 - 49) = √576 = 24.
sin θ = P/H = 24/25.
Question 25: Tan and Cot
If tan A = cot B, then A + B = ?
A = 90 - B
A + B = 90°.
