Double Angle Formulas

Double-angle formulas can be expanded to multiple-angle functions (triple, quadruple, quintuple, and so on) by using the angle sum formulas, and then reapplying the double-angle formulas.

$\large sin(A+B)=sinA\;cosB+cosA\;sinB$

$\large sin(A-B)=sinA\;cosB-cosA\;sinB$

$\large cos(A+B)=cosA\;cosB-sinA\;sinB$

$\large cos(A-B)=cosA\;cosB+sinA\;sinB$

$\large sin\alpha +sin\beta =2sin\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}$

$\large sin\alpha -sin\beta =2sin\frac{\alpha -\beta }{2}cos\frac{\alpha +\beta }{2}$

$\large cos\alpha +cos\beta =2cos\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}$

$\large cos\alpha -cos\beta =-2sin\frac{\alpha +\beta }{2}sin\frac{\alpha -\beta }{2}$

$\large sin2\alpha =2\;sin\alpha\;cos\alpha$

$\large cos2\alpha =cos^{2}\alpha -sin^{2}\alpha = 2cos^{2}\alpha -1=1-2sin^{2}\alpha$

$\large tan2\alpha =\frac{2tan\alpha }{1-tan^{2}\alpha }$

Half Angle Formulas

$\large sin\left ( \frac{a}{2} \right )=\pm \sqrt{\frac{(1-cos\;a)}{2}}$

$\large cos\left ( \frac{a}{2} \right )=\pm \sqrt{\frac{(1+cos\;a)}{2}}$

$\large tan\left ( \frac{a}{2} \right )=\frac{1-cos\;a}{sin\;a}=\frac{sin\;a}{1+cos\;a}$

$\large sin\left ( \frac{a}{2} \right )=\sqrt{\frac{1-cos(a)}{2}}$

$\large cos\left ( \frac{a}{2} \right )=\sqrt{\frac{1+cos(a)}{2}}$

$\large tan\left ( \frac{a}{2} \right )=\sqrt{\frac{1-cos(a)}{1+cos(a)}}$

Practise This Question

When Ted entered the class, Mr. McMurphy was teaching integers!! Ted didn’t even know what was going on in the class before, but, the moment he heard the word ‘integer’ from his mouth, he became super attentive. Suddenly, Mr. McMurphy started asking questions.

He said: “If you add or subtract two integers, the result will always be an integer. What is this property called?” Thinking that anytime he could ask him, Ted started looking at the book for the answer. To his relief, one of his classmates gave right answer. The right answer was -