5. $co{s}^{-1}\left(-\frac{1}{2}\right)$

<p>Let cos ^-1<span title="Click to copy mathml"><math><mrow><mrow><mo> (</mo><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>&nbsp;=y Then cos y =&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>&nbsp;= − cos <span title="Click to copy mathml"><br><math><mrow><mfrac><mrow><mn></mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span><math><mrow><mfrac><mrow></mrow><mrow></mrow></mfrac></mrow></math>&nbsp;cos&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mo> (</mo><mrow><mi>π</mi><mo>−</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p><p>= cos&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>3</mn><mi>π</mi><mo>−</mo><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span></p><p>= cos&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span></p><p><math></math>&nbsp; (E) We know that the range of principal value</p><p>branch of cos⁻¹ is [0, <span title="Click to copy mathml"><math><mrow><mfrac><mrow><mi>π</mi></mrow></mfrac></mrow></math></span>] and cos <span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span></p><p>Principal value of cos−1&nbsp;<span title="Click to copy mathml"><math><mrow><mrow><mo> (</mo><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math></span>&nbsp;is&nbsp;<span title="Click to copy mathml"><math><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></span></p>

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