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The value of cos⁻1(cos 5π/3) + sin⁻1 (sin 5π/3) is
- a)π/2
- b)5π/3
- c)10π/3
- d)0
Correct answer is option 'D'. Can you explain this answer?
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The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/...
(cos 9 + cos 13 + cos 17 + ... + cos 177)
We can start by using the identity:
cos A cos B = 1/2 (cos(A+B) + cos(A-B))
Let A = 1 and B = 5:
cos 1 cos 5 = 1/2 (cos 6 + cos 4)
Next, we can use the identity:
cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
Let A = 6 and B = 4:
cos 6 + cos 4 = 2 cos 5 cos 1
Substituting into the original expression:
cos1(cos 5(cos 9 + cos 13 + cos 17 + ... + cos 177))
= 1/2 (cos 6 + cos 4) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (cos 5 cos 1) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) (cos 6 + cos 4) (2 cos 5 cos 1) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) (cos 6 + cos 4) (cos 5 + cos 7) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) [(cos 6 cos 5 cos 9 + cos 6 cos 5 cos 13 + ... + cos 6 cos 5 cos 177) + (cos 4 cos 5 cos 9 + cos 4 cos 5 cos 13 + ... + cos 4 cos 5 cos 177) + (cos 6 cos 7 cos 9 + cos 6 cos 7 cos 13 + ... + cos 6 cos 7 cos 177) + (cos 4 cos 7 cos 9 + cos 4 cos 7 cos 13 + ... + cos 4 cos 7 cos 177)]
We can simplify each term using the identity cos A cos B cos C = 1/4 [cos(A+B+C) + cos(A-B+C) + cos(-A+B+C) + cos(-A-B+C)]:
cos 6 cos 5 cos 9 + cos 6 cos 5 cos 13 + ... + cos 6 cos 5 cos 177
= 1/4 [cos(20) + cos(8) + cos(-4) + cos(-16) + ... + cos(-340)]
cos 4 cos 5 cos 9 + cos 4 cos 5 cos 13 + ... + cos 4 cos 5 cos 177
= 1/4 [cos(18) + cos(2) + cos(-14) + cos(-30) + ... + cos(-338)]
cos 6 cos 7 cos 9 + cos 6 cos 7 cos 13 + ... + cos 6 cos 7 cos 177
= 1/4 [cos(22) + cos(10) + cos(-2) + cos(-14) + ... + cos(-338)]
cos 4 cos 7 cos 9
We can start by using the identity:
cos A cos B = 1/2 (cos(A+B) + cos(A-B))
Let A = 1 and B = 5:
cos 1 cos 5 = 1/2 (cos 6 + cos 4)
Next, we can use the identity:
cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
Let A = 6 and B = 4:
cos 6 + cos 4 = 2 cos 5 cos 1
Substituting into the original expression:
cos1(cos 5(cos 9 + cos 13 + cos 17 + ... + cos 177))
= 1/2 (cos 6 + cos 4) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (cos 5 cos 1) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) (cos 6 + cos 4) (2 cos 5 cos 1) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) (cos 6 + cos 4) (cos 5 + cos 7) (cos 9 + cos 13 + cos 17 + ... + cos 177)
= (1/2) [(cos 6 cos 5 cos 9 + cos 6 cos 5 cos 13 + ... + cos 6 cos 5 cos 177) + (cos 4 cos 5 cos 9 + cos 4 cos 5 cos 13 + ... + cos 4 cos 5 cos 177) + (cos 6 cos 7 cos 9 + cos 6 cos 7 cos 13 + ... + cos 6 cos 7 cos 177) + (cos 4 cos 7 cos 9 + cos 4 cos 7 cos 13 + ... + cos 4 cos 7 cos 177)]
We can simplify each term using the identity cos A cos B cos C = 1/4 [cos(A+B+C) + cos(A-B+C) + cos(-A+B+C) + cos(-A-B+C)]:
cos 6 cos 5 cos 9 + cos 6 cos 5 cos 13 + ... + cos 6 cos 5 cos 177
= 1/4 [cos(20) + cos(8) + cos(-4) + cos(-16) + ... + cos(-340)]
cos 4 cos 5 cos 9 + cos 4 cos 5 cos 13 + ... + cos 4 cos 5 cos 177
= 1/4 [cos(18) + cos(2) + cos(-14) + cos(-30) + ... + cos(-338)]
cos 6 cos 7 cos 9 + cos 6 cos 7 cos 13 + ... + cos 6 cos 7 cos 177
= 1/4 [cos(22) + cos(10) + cos(-2) + cos(-14) + ... + cos(-338)]
cos 4 cos 7 cos 9
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The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer?
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The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer?.
The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The value of cos1(cos 5π/3) + sin1 (sin 5π/3) isa)π/2b)5π/3c)10π/3d)0Correct answer is option 'D'. Can you explain this answer?.
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