Sin420°+Cos390°-Cos(-300°)Sin(-330°) Solve And Answer Value?

3 Answers
Apr 28, 2018

sqrt3-3/4

Explanation:

sin(420^o)+cos390^o-cos(-300^o)sin(-330^o)

=sin(420^o)+cos390^o-cos(300^o)(-sin(330))

=sin(420^o)+cos(390^o)+cos(300^o)sin(330^o)

=sin(5xx90^o-30^o)+cos(5xx90^o-60^o)+cos(4xx90^o- 60^o)sin(4xx90^o-30^o)

=cos30^0+sin60^0+(sin60^o)(-cos30^o)

=sqrt3/2+sqrt3/2-sqrt3/2.sqrt3/2

=sqrt3-3/4

Apr 28, 2018

Adding or subtracting 360^circ each angle brings them in range, where the form is recognized to be the sine difference angle formula, and works out to 1/2.

Explanation:

Angles that differ by a multiple of 360^circ are called coterminal , and have the same value for their trig functions.

Let's get all of these in the range -180^circ to 180^circ.

sin 420^circ cos 390^circ - cos(-300^circ) sin (-330^circ)

= sin (420^circ -360^circ) cos (390^circ-360^circ) - cos(-300^circ+360^circ) sin (-330^circ+360^circ)

= sin (60^circ) cos (30^circ) - cos(60^circ) sin (30^circ)

We know what all of those are so we could work this out, or recognize it as the sine difference angle formula:

sin(a-b) = sina cos b - cos a sin b

= sin (60^circ) cos (30^circ) - cos(60^circ) sin (30^circ)

= sin (60^circ - 30^circ) = sin 30^circ = 1/2

Apr 28, 2018

[Sin420°Cos390°]-[Cos(-300°)Sin(-330°)]

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"As "[ color(magenta)(cos(-x) = cosx ; sin(-x)= -sinx)]

=>Sin420°Cos390°+Cos300°Sin330°

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Now split the angle measurement in terms of 360^@

=>Sin(360+60)°Cos(360+30)°+Cos(360-60)°Sin(360-30)°

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=>Sin60°Cos30°- Cos60°Sin30°

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This is of the form, color(red)(SinACosB- CosASinB = sin(A-B)

=>sin30^@ =1/2