How to calculateĀ the expression sin105*cos45-sin45*cos105?

Expert Answers
justaguide eNotes educator| Certified Educator

We have to evaluate sin 105*cos 45 - sin 45*cos 105

As can be seen the expression is of the form sin x*cos y - sin y*cos x which is equal to sin (x - y)

sin 105*cos 45 - sin 45*cos 105

=> sin (105 - 45)

=> sin 60

=> (sqrt 3)/2

The required value of the expression is (sqrt 3)/2

chaobas | Student

We have the formula

(sin a * cos b)-(sin b * cos a) = sin (a-b) : were a and b are the angle

Here in the equation

sin105*cos45-sin45*cos105, so if we compare this one with the above equation, we4 can see that a=105 and b = 45

So

sin105*cos45-sin45*cos105 = sin (105-45) = sin 60

and the value of sin 60 is (sqrt 3)/2

giorgiana1976 | Student

To evaluate the given expression, we'll use the identity:

sin(x-y)=sin x*cos y - sin y*cos y

Let x = 105 and y = 45

We'll re-write the given difference:

sin 105*cos 45 - sin 45*cos 105 = sin (105 - 45)

sin 105*cos 45 - sin 45*cos 105 = sin 60

But sin 60 = (sqrt 3)/2

Therefore, the value of the given expression is sin 105*cos 45 - sin 45*cos 105 = (sqrt 3)/2.

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