Why isn't sin 30+ sin 60= sin 90?

Answer 1

The sine function is not a linear function.

The linear addition property is

F(a+b) = F(a) + F(b)

However sine has another formula for addition

The addition formula for sine is

Sin(A+B) = Sin(A)Cos(B) + Sin(B)Cos(A)

So Sin(90) is equal to this

Sin(30+60) = Sin(90)
Sin(30+60) = Sin(30)Cos(60) + Sin(60)Cos(30)
= (.5)(.5)+ ((sqrt(3))/2)((sqrt(3))/2)
= 1

Here is a link to the proof of the Sine Addition formula

http://www.themathpage.com/aTrig/sum-proof.htm

Other common Trigonometry Addition/Difference formulas
are

Sine Difference formula

Sin(A-B) = Sin(A)Cos(B) - Sin(B)Cos(A)

Cosine Addition formula

Cos(A+B) = Cos(A)Cos(B) - Sin(A)Sin(B)

Cosine Difference formula

Cos(A-B) = Cos(A)Cos(B) + Sin(A)Sin(B)

Answer 2

1/2 + √3/2 ≠ 1

Answer 3

thats coz 30 and 60 are angles, and not values which can be added and subtracted.
sin30 = 1/2
sin60 = 1.732/2 (I couldn't type root 3 on keyboard)

But sin 90 = 1.

0.5 + 1.732/2 is not equal to 1.

That answers your problem.

Answer 4

b/c thers a proper for a sin of sum of angles
is is sin(a+b)=sin(a)cos(a)+sin(b)cos(b)
sin(a-b)=sin(a)cos(a)-sin(b)cos(b)