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ACT Exam  >  ACT Questions  >  If tan 48° tan 23° tan 42° tan 67... Start Learning for Free
If tan 48° tan 23° tan 42° tan 67° = tan(A + 30°) then A will be
  • a)
    30∘
  • b)
    15∘
  • c)
    45∘
  • d)
    60∘
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If tan 48° tan 23° tan 42° tan 67° = tan(A + 30°) ...
Formula used:
tan(90° - θ) = cot θ
tan θ × cot θ = 1
Calculation:
Given that,
tan 48° tan 23° tan 42° tan 67° = tan(A + 30∘)
⇒ tan (90° - 42°)tan(90 - 67°)tan 42°tan 67° = tan(A +  30°)
⇒ cot 42° cot 67° tan 42°tan 67° = tan(A + 30°)
∵ tan θ × cot θ = 1
⇒ 1 × 1 = tan(A + 30°)
⇒ tan 45° = tan(A + 30°)    (∵ tan 45° = 1)
⇒ 45° = A + 30°
⇒ A = 15°
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Most Upvoted Answer
If tan 48° tan 23° tan 42° tan 67° = tan(A + 30°) ...
Understanding the Problem
To solve the equation tan 48° tan 23° tan 42° tan 67° = tan(A + 30°), we need to simplify the left-hand side and explore relationships between the angles.
Using Angle Identities
1. Complementary Angles:
- We know that tan(90° - x) = cot(x).
- Therefore, tan 67° = cot 23° and tan 42° = cot 48°.
2. Rewriting the Expression:
- This gives us:
- tan 48° tan 23° tan 42° tan 67° = tan 48° cot 48° tan 23° cot 23°.
- Both tan 48° and cot 48° multiply to 1, and the same for tan 23° and cot 23°:
- Thus, tan 48° cot 48° = 1 and tan 23° cot 23° = 1.
3. Final Simplification:
- Therefore, tan 48° tan 23° tan 42° tan 67° = 1.
Equating Both Sides
Now we have 1 = tan(A + 30°).
Finding A
1. Using the tan Identity:
- The tangent of an angle is 1 when the angle is 45° (plus any integer multiple of 180°).
- Therefore, A + 30° = 45°.
2. Solving for A:
- Thus, A = 45° - 30° = 15°.
Conclusion
The value of A is 15°, confirming that the correct answer is option 'B'. This demonstrates how understanding complementary angles and the properties of tangent can simplify the problem effectively.
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Community Answer
If tan 48° tan 23° tan 42° tan 67° = tan(A + 30°) ...
Formula used:
tan(90° - θ) = cot θ
tan θ × cot θ = 1
Calculation:
Given that,
tan 48° tan 23° tan 42° tan 67° = tan(A + 30∘)
⇒ tan (90° - 42°)tan(90 - 67°)tan 42°tan 67° = tan(A +  30°)
⇒ cot 42° cot 67° tan 42°tan 67° = tan(A + 30°)
∵ tan θ × cot θ = 1
⇒ 1 × 1 = tan(A + 30°)
⇒ tan 45° = tan(A + 30°)    (∵ tan 45° = 1)
⇒ 45° = A + 30°
⇒ A = 15°
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If tan 48° tan 23° tan 42° tan 67° = tan(A + 30°) then A will bea)30b)15c)45d)60Correct answer is option 'B'. Can you explain this answer?
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