Question #e3aad

Question

1614 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-16T15:33:02

We are to evalute $sin33$ .

$sin33=sin(18+15)$

$=sin18cos15+cos18sin15.....(1)$

Evaluation of $sin18 and cos18$

Let $A=18^@$

$=>5A=90^@$

$=>3A=90^@-2A$

$:.cos(3A)=cos(90^@-2A)$

$=>4cos^3A-3cosA=sin2A=2sinAcosA$

$=>4cos^2A-3=2sinA$

$=>4-4sin^2A-3=2sinA$

$=>4sin^2+2sinA-1=0$

$=>sinA=(-2+sqrt(2^2-4*4* (-1)))/(2*4)$

$=(-2+sqrt20)/8=(sqrt5-1)/4$

$:.sin(18^@)=(sqrt5-1)/4$ ,

$cos18^@=sqrt(1-sin^2 18^@)$

$=sqrt(1-(sqrt5-1)^2/16)$

$=sqrt(16-5-1+2sqrt5)/4$

$=1/4sqrt(10+2sqrt5)$

Evaluation of $sin15 and cos15$

$sin15=sqrt(1/2(1-cos30))$

$=sqrt(1/2(1-sqrt3/2))$

$=sqrt(1/8(4-2sqrt3))$

$=sqrt(1/8(sqrt3-1)^2)$

$=(sqrt3-1)/(2sqrt2)$

$cos15=sqrt(1/2(1+cos30))$

$=sqrt(1/2(1+sqrt3/2))$

$=sqrt(1/8(4+2sqrt3))$

$=sqrt(1/8(sqrt3+1)^2)$

$=(sqrt3+1)/(2sqrt2)$

Using relation (1)

$sin33=sin18cos15+cos18sin15$

$=(sqrt5-1)/4*(sqrt3+1)/(2sqrt2)+sqrt(10+2sqrt5)/4*(sqrt3-1)/(2sqrt2)$

$=1/(8sqrt2)((sqrt5-1)(sqrt3+1)+(sqrt(10+2sqrt5))(sqrt3-1))$