What is the principal value of an inverse trigonometric function?

<p>To understand this, Assume you have a bucket that has infinite number of apples and if your mother asks "give me the apple". How will you figure out which one is "The Apple", she is asking for.</p><p>Similarly any inversre trigonometric functions behaves like a&nbsp;<em>Many-one Function; </em>which means, </p><p>For Example <math ><semantics><mrow><msup><mrow><mi>sin</mi><mo>? </mo></mrow><mrow><mo>? </mo><mn>1</mn></mrow></msup><mrow><mo fence="true"> (</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow>&lt;annotation encoding="application/x-tex"&gt;\sin^ {-1}\left (\frac {2} {3}\right)</annotation></semantics></math> can have many solutions, we need to fix one solutions which can be used as standerd value for the function.</p><ul><li>A standerd value (Angles) of any inverse trigonometric value lies between a fiexed range is known as principal value.&nbsp;</li></ul><p>For Ex; The value of&nbsp;<math ><semantics><mrow><msup><mrow><mi>sin</mi><mo>? </mo></mrow><mrow><mo>? </mo><mn>1</mn></mrow></msup><mrow><mo fence="true"> (</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow>&lt;annotation encoding="application/x-tex"&gt;\sin^ {-1}\left (\frac {2} {3}\right)</annotation></semantics></math> will always lie between –? /2 to? /2.</p><p><br><span class="mathml" contenteditable="false"> <math > <mrow> <mi>y</mi> <mo>=</mo> <msup> <mi>sin</mi> <mo>?1</mo> </msup> <mo> (</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>? </mo> <mi>sin</mi> <mo> (</mo> <mi>y</mi> <mo>)</mo> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow> <mo>? </mo> <mi>y</mi> <mo>? </mo> <mo> [</mo> <mo>? </mo> <mfrac> <mi>? </mi> <mn>2</mn> </mfrac> <mo>, </mo> <mfrac> <mi>? </mi> <mn>2</mn> </mfrac> <mo>]</mo> </mrow> </math> </span>&nbsp;</p>

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