How do you find the exact value of cos[2arcsin(35)arctan(512)]?

2 Answers
Mar 28, 2016

Possible values are ±0.1108 or ±0.6277

Explanation:

arcsin(35)=x means sinx=(35)=0.6.

As sinx=0.6 for x=36.87o and sine is negative in third and fourth quadrant, x=18036.87o or 216.87o and x=360o36.87o=323.13o.

arctan(512)=x means tanx=(512).

As tanx=512 for x=22.62o and tan is positive in first and third quadrant, x=22.62o or x=180o+22.62o.or 202.62o.

Hence cos{2arcsin(35)arctan(512)]=cos[2×216.87o22.62o]=cos411.12o=cos51.12o=0.6277 or

cos{2arcsin(35)arctan(512)]=cos[2×216.87o202.62o]=cos411.12o=cos231.12o=0.6277 or

cos{2arcsin(35)arctan(512)]=cos[2×323.13o22.62o]=cos623.64o=0.1108 or

cos{2arcsin(35)arctan(512)]=cos[2×323.13o202.62o]=cos443.64o=0.1108

Mar 28, 2016

Values in exactitude are ±12125 and ±68125.

Explanation:

Let A=arcsin(35) and B=arctan(512).
Then, sinA=35,cosA=±45.

cos2A=12sin2A=725

sin2A=2sinAcosA=2425and2425, respectively..

Also, tanB=512,(sinB=513,cosB=1213)and(sinB=513,cosB=1213)

The given expression is

cos(2A-B) = cos 2A cos B+sin 2A sin B

= ±84±120375.

Note that both sin B and cos B have the same sign + or .