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Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)?
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Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)?
Given Equation:
Sec θ – tan θ = 1/3
To find:
Value of (sec θ + tan θ)
Step 1: Rewrite the given equation
Sec θ - tan θ = 1/3
Sec θ = tan θ + 1/3
Step 2: Use trigonometric identities
Sec θ = 1/cos θ
Tan θ = sin θ / cos θ
Step 3: Substitute the trigonometric identities into the equation
1/cos θ = sin θ / cos θ + 1/3
1 = sin θ + cos θ / 3cos θ
3cos θ = sin θ + cos θ
Step 4: Square both sides of the equation
(3cos θ)^2 = (sin θ + cos θ)^2
9cos^2 θ = sin^2 θ + 2sin θ cos θ + cos^2 θ
Step 5: Use Pythagorean identity
1 = sin^2 θ + cos^2 θ
Substitute this into the equation:
9cos^2 θ = 1 + 2sin θ cos θ
9cos^2 θ = 1 + sin 2θ
Step 6: Simplify the equation
cos^2 θ = 1/9
cos θ = ±1/3
Step 7: Find sec θ and tan θ
Since cos θ = 1/3, we have:
Sec θ = 1/cos θ = 3
Tan θ = sin θ / cos θ = √8 / 3
Step 8: Find sec θ + tan θ
Sec θ + tan θ = 3 + √8 / 3
Sec θ + tan θ = (3√8 + 8) / 3
Sec θ + tan θ = (3√8 + 8) / 3
Therefore, the value of (sec θ + tan θ) is (3√8 + 8) / 3.
Given Equation:
Sec θ – tan θ = 1/3
To find:
Value of (sec θ + tan θ)
Step 1: Rewrite the given equation
Sec θ - tan θ = 1/3
Sec θ = tan θ + 1/3
Step 2: Use trigonometric identities
Sec θ = 1/cos θ
Tan θ = sin θ / cos θ
Step 3: Substitute the trigonometric identities into the equation
1/cos θ = sin θ / cos θ + 1/3
1 = sin θ + cos θ / 3cos θ
3cos θ = sin θ + cos θ
Step 4: Square both sides of the equation
(3cos θ)^2 = (sin θ + cos θ)^2
9cos^2 θ = sin^2 θ + 2sin θ cos θ + cos^2 θ
Step 5: Use Pythagorean identity
1 = sin^2 θ + cos^2 θ
Substitute this into the equation:
9cos^2 θ = 1 + 2sin θ cos θ
9cos^2 θ = 1 + sin 2θ
Step 6: Simplify the equation
cos^2 θ = 1/9
cos θ = ±1/3
Step 7: Find sec θ and tan θ
Since cos θ = 1/3, we have:
Sec θ = 1/cos θ = 3
Tan θ = sin θ / cos θ = √8 / 3
Step 8: Find sec θ + tan θ
Sec θ + tan θ = 3 + √8 / 3
Sec θ + tan θ = (3√8 + 8) / 3
Sec θ + tan θ = (3√8 + 8) / 3
Therefore, the value of (sec θ + tan θ) is (3√8 + 8) / 3.
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Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)?
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Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)?.
Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Sec θ – tan θ = 1/3, then find the value of (sec θ + tan θ)?.
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