How do you evaluate Sin (165)?

1 Answer
George C.
Sep 29, 2016

sin 165^@ = 1/4(sqrt(6)-sqrt(2))

Explanation:

Some things we will use:

sin(theta) = sin (180^@ - theta)

sin(alpha-beta) = sin alpha cos beta - sin beta cos alpha

sin 45^@ = cos 45^@ = sqrt(2)/2

sin 30^@ = 1/2

cos 30^@ = sqrt(3)/2

Hence we find:

sin 165^@ = sin (180^@-165^@)

color(white)(sin 165^@) = sin 15^@

color(white)(sin 165^@) = sin (45^@ - 30^@)

color(white)(sin 165^@) = sin 45^@ cos 30^@ - sin 30^@ cos 45^@

color(white)(sin 165^@) = sqrt(2)/2 sqrt(3)/2 - 1/2 sqrt(2)/2

color(white)(sin 165^@) = 1/4(sqrt(6)-sqrt(2))

color(white)()
Footnotes

The trigonometric values we used in our derivation can be observed in the following right angled triangles:

enter image source here

Hence sin 45^@ = cos 45^@ = 1/sqrt(2) = sqrt(2)/2

enter image source here

Hence sin 30^@ = 1/2 and cos 30^@ = sqrt(3)/2

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