`105^@ = 60^@ + 45^@` Find the exact values of the sine, cosine, and tangent of the angle.
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You need to evaluate the sine of `105^o` , using the formula `sin(a+b) = sin a*cos b + sin b*cos a` such that:
`sin(105^o) = sin(60^o + 45^o) = sin 60^o*cos 45^o + sin 45^o*cos 60^o`
`sin(105^o) = (sqrt3)/2*(sqrt2)/2 + (sqrt2)/2*1/2`
`sin(105^o) = (sqrt2)/2*(sqrt3 + 1)/2`
You need to evaluate the cosine of `105^o` , using the formula `cos(a+b) = cos a*cos b - sin b*sin a` such that:
`cos (105^o) = cos (60^o + 45^o) = cos 60^o*cos 45^o - sin 45^o*sin 60^o`
`cos (105^o) = 1/2*(sqrt2)/2 - (sqrt2)/2*(sqrt3)/2`
`cos (105^o) = (sqrt2)/2*(1 - sqrt3)/2`
You need to evaluate the tangent of `105^o` , such that:
`tan 105^o = (sin(105^o))/(cos (105^o))`
`tan 105^o = ((sqrt2)/2*(sqrt3 + 1)/2)/((sqrt2)/2*(1 - sqrt3)/2)`
`tan 105^o = (sqrt3 + 1)/(1 - sqrt3)`
`tan 105^o = ((sqrt3 + 1)*(1 + sqrt3)/(1 - 3)`
`tan 105^o = -((sqrt3 + 1)^2)/2`
Hence, evaluating the sine, cosine and tangent of `105^o` , yields `sin(105^o) = (sqrt2)/2*(sqrt3 + 1)/2, cos (105^o) = (sqrt2)/2*(1 - sqrt3)/2, tan 105^o = -((sqrt3 + 1)^2)/2.`
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