Question

Find the exact value, without a

calculator.
30°
sin 15° sin
tan 15º =
2
cos 15° 30°
2
COS
II
[ ? ] - [
] + [
Enter

Answer 1

`sin\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1-cos(\theta)}{2}} \qquad cos\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1+cos(\theta)}{2}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ sin\left( \cfrac{30^o}{2} \right)\implies \sqrt{\cfrac{1-cos(30^o)}{2}}\implies \sqrt{\cfrac{1-(√(3))/(2)}{2}}\implies \sqrt{\cfrac{~~(2-√(3))/(2)~~}{2}}`

`\sqrt{\cfrac{2-√(3)}{2}\cdot \cfrac{1}{2}}\implies \sqrt{\cfrac{2-√(3)}{4}}\implies \boxed{\cfrac{\sqrt{2-√(3)}}{2}} \\\\[-0.35em] ~\dotfill\\\\ cos\left( \cfrac{30^o}{2} \right)\implies \sqrt{\cfrac{1+cos(30^o)}{2}}\implies \sqrt{\cfrac{1+(√(3))/(2)}{2}}\implies \sqrt{\cfrac{~~(2+√(3))/(2)~~}{2}} \\\\\\ \sqrt{\cfrac{2+√(3)}{2}\cdot \cfrac{1}{2}}\implies \sqrt{\cfrac{2+√(3)}{4}}\implies \boxed{\cfrac{\sqrt{2+√(3)}}{2}} \\\\[-0.35em] ~\dotfill`

`\cfrac{~~ sin\left( (30^o)/(2) \right)~~}{cos\left( (30^o)/(2) \right)}\implies \cfrac{~~\frac{\sqrt{2-√(3)}}{2} ~~}{\frac{\sqrt{2+√(3)}}{2}}\implies \cfrac{\sqrt{2-√(3)}}{2}\cdot \cfrac{2}{\sqrt{2+√(3)}} \\\\\\ \cfrac{\sqrt{2-√(3)}}{\sqrt{2+√(3)}}\implies \blacktriangleright \sqrt{\cfrac{2-√(3)}{2+√(3)}} \blacktriangleleft`