Here you will learn what is the formula of sin 2A in terms of sin and cos and also in terms of tan with proof and examples.<\/p>\n
Let’s begin –<\/p>\n
Sin 2A = 2 sin A cos A<\/p><\/blockquote>\n
Proof :<\/strong><\/p>\n
We have,<\/p>\n
Sin (A + B) = sin A cos B + cos A sin B<\/p>\n
Replacing B by A,<\/p>\n
\\(\\implies\\) sin 2A = sin A cos A + cos A sin A<\/p>\n
\\(\\implies\\) sin 2A = 2 sin A cos A<\/p>\n
We can also write above relation in terms of angle A\/2, just replace A by A\/2, we get<\/p>\n
sin A = \\(2 sin ({A\\over 2}) cos ({A\\over 2})\\)<\/p><\/blockquote>\n
(ii) Sin 2A Formula in Terms of Tan :<\/h3>\n
Sin 2A = \\(2 tan A\\over 1 + tan^2 A\\)<\/p><\/blockquote>\n
Proof :<\/strong><\/p>\n
We have,<\/p>\n
sin 2A = 2 sin A cos A<\/p>\n
\\(\\implies\\) sin 2A = \\(2 sin A cos A\\over sin^2 A + cos^2 A\\)<\/p>\n
[ \\(\\because\\) \\(sin^2 A + cos^2 A\\) = 1 ]<\/p>\n
Now, Dividing numerator and denominator by \\(cos^2 A\\),<\/p>\n
\\(\\implies\\) sin 2A = \\({2sin A cos A\\over cos^2 A}\\over {sin^2 A + cos^2 A\\over cos^2 A}\\)<\/p>\n
\\(\\implies\\) sin 2A = \\(2 tan A\\over 1 + tan^2 A\\)<\/p>\n
We can also write above relation in terms of angle A\/2, just replace A by A\/2, we get<\/p>\n
sin A = \\(2 tan ({A\\over 2})\\over 1 + tan^2 ({A\\over 2})\\)<\/p><\/blockquote>\n