A student incorrectly wanted to show that sin (A+B) = sin A + sin B. They made their attempt at proving this false claim by letting A= 120 ^(∘) and B=240^(∘).
The student's reasoning is incorrect because trigonometric functions are not additive.
sin (A+B) * ≠ sin A + sin BTo add them, we need to use the Angle Sum Identity for sine.
sin ( A+ B) = sin A cos B + cos A sin B
Let's see how this identity works with the values of A and B chosen by the student.
Therefore, the Angle Sum Identity give us the same result as the student's method, but there are plenty of examples where the addition does not work. Let's see one of them!
sin ( 30^(∘)+ 60^(∘)) = sin 90 ^(∘)
We will calculate the exact value of the previous trigonometric expression using the calculator, the student's method, and the Angle Sum Identity.
Procedure
Result
Calculator
sin 90 ^(∘)
1
Student's Method
sin 30^(∘)+ sin 60^(∘)
≈ 1.37
Angle Sum Identity
sin 30^(∘) cos 60^(∘) + cos 30^(∘) sin 60^(∘)
1
As we can see, the student's method gives us a different answer. This shows us that the trigonometric functions are not additive.
