Ex. 34. Statement I: x = k π 2 , k ∈ I does not represent the general solution of trigonometric equation. Statement II: Both x = n π , n ∈ I and x = k π 2 , k ∈ I satisfy the trigonometric equation sin 13 x − sin 13 x cos 2 x = 0 .

Solution: sin 13 x ( 1 − cos 2 x ) = 0 ⇒ sin 13 x = 0 or cos 2 x = 1 ⇒ 13 x = k π or cos 2 x = 1 ⇒ x = k π 13 , k ∈ I ⇒ 2 x = 2 m π ⇒ x = m π , n ∈ I ⇒ Statement I is false and Statement II is true. ∴ Answer is (d).

Trigonometry & Inverse Trigonometry

Trigonometric Equations

Grade 11

Question:

Ex. 34. Statement I: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>π</mi></mrow><mn>2</mn></mfrac></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math> does not represent the general solution of trigonometric equation.Statement II: Both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>π</mi></mrow><mn>2</mn></mfrac></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math> satisfy the trigonometric equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sin</mi><mn>13</mn><mi>x</mi><mo>−</mo><mi>sin</mi><mn>13</mn><mi>x</mi><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math>.

(a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

(b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.

(c) Statement I is true, Statement II is false.

(d) Statement I is false, Statement II is true.

Step-by-Step Solution

Key Concept: Solve the trigonometric equation by factoring and find all solutions to verify which statements are true.

Solution:<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sin</mi><mn>13</mn><mi>x</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>cos</mi><mn>2</mn><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sin</mi><mn>13</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>13</mn><mi>x</mi><mo>=</mo><mi>k</mi><mi>π</mi></mrow></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>π</mi></mrow><mn>13</mn></mfrac></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>m</mi><mi>π</mi></mrow></math> ⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mi>m</mi><mi>π</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math>⇒ Statement I is false and Statement II is true.∴ Answer is (d).

Correct Answer: D

Question:

Step-by-Step Solution

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