Question

Find sin A.
If sin(60-A) sin(60+A)​

Answer 1

Answer:

To calculate this, we can use the trigonometric identity

SinC- sinD=2cos((c+d)/2)sin((c-d)/2)

So , now using the above identity

Sin(a+60)-sin(a-60)

=2cos((a+60+a-60)/2)sin((a+60-(a-60))/2)

=2cos(2a/2)sin((a+60-a+60)/2)

=2cos(a)sin(120/2)

=2cos(a)sin60

=2cos(a)(3^1/2)/2

=1.73cos(a)

Answer 2

Use Identity :-

sin(A+B)

= sinAcosB - cosAsinB

So,

sin(A+60)

= sinAcos60 - sin60cosA

Substituting 60 as -60 and noting that cos(-X) = cos(X) due its symmetric nature,

sin(A-60)

= sinAcos60 + sin60cosA

Hence, using these expansions you'll get the following:

sin(A+60) - sin(A-60)

= sinAcos60 - sin60cosA - (sinAcos60 + sin60cosA)

Taking the right hand side and simplifying it by eliminating the parentheses and rearranging:

sinAcos60 -sinAcos60 - sin60cosA - sin60cosA

= -2sin60cosA

sin60 is 0.5 X square root of 3

Therefore,

-2sin60cosA = - 2 * 0.5 * root3 *cosA

All of it basically sums up to -(root3)*cosA

The answer makes sense because you're treating A as a variable, and the graph doesn't cancel out like more aesthetic examples, so at a particular value of A the the graph of your function would map it's domain to the value of -(root3)*cosA