Find the six trigonometric functions at theta pi3 Verify th

Find the six trigonometric functions at theta = pi/3 Verify the identity: sin theta/1 + cos theta + cos theta/1 - cos theta = Csc theta cosx/1 - sinx = secx + tanx 1 + cos x/sinx + sinx + sinx/1 + cosx = 2cscx Find the solutions of the equations in the interval[0, 2pi] xsinxcosx = 0

Solution

2)=_/3

sin(_/3) =(3)_/2

cos(_/3) =1_/2

tan(_/3) =3

csc(_/3) =2_/3

sec(_/3) =2  

cot(_/3) =1_/3

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a)sin_/(1+cos) +cos_/(1-cos)

=[sin(1-cos) +cos(1+cos)]_{/(1+cos)(1-cos)}

=[sin-sincos +cos+cos²)]_/(1-cos²)

=[sin-sincos +cos+cos²)]_/(sin²)

not an identity

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b)cosx_/(1-sinx)

=[cosx_/(1-sinx)]*[(1+sinx)_/(1+sinx)]

=cosx(1+sinx)_/(1-sin²x)

=cosx(1+sinx)_/(cos²x)

=(1+sinx)_/(cosx)

=(1_/cosx)+(sinx_/cosx)

=secx +tanx

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c)[(1+cosx)_/sinx]*[sinx_/(1+cosx)]

=[(1+cosx)²+(sinx)²]_{/[sinx(1+cosx)]}

=[1+2cosx+(cosx)²+(sinx)²]_{/[sinx(1+cosx)]}

=[1+2cosx+1]_{/[sinx(1+cosx)]}

=[2+2cosx]_{/[sinx(1+cosx)]}

=2[1+cosx]_{/[sinx(1+cosx)]}

=2_/sinx

=2cscx