The sine of the sum and difference of two angles is a fundamental trigonometric identity that is used in many areas of mathematics and physics. The formula, also known as the sine double angle formula, is given by:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
sin(a – b) = sin(a)cos(b) – cos(a)sin(b)
where a and b are the angles being considered.
This identity is derived from the addition formula for sine, which states that:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
To prove the above formula, we can use the complex exponential representation of sine and cosine:
e^(i(a+b)) = e^(ia)e^(ib)
sin(a+b) = (e^(i(a+b))e^(i(a+b)))/(2i)
= (e^(ia)e^(ib) – e^(ia)e^(ib))/(2i)
= (e^(ia)(e^(ib)e^(ib)) + e^(ia)(e^(ib) – e^(ib)))/(2i)
= (e^(ia)2i sin(b) + e^(ia)2i sin(b))/(2i)
= sin(a)cos(b) + cos(a)sin(b)
The same process can be used to derive the formula for sin(a – b).
The sine double angle formula is useful in many areas of mathematics and physics. For example, in physics, it is used to calculate the motion of a simple harmonic oscillator, such as a mass on a spring. In engineering, it is used to calculate the power output of a threephase alternating current system. In trigonometry, it is used to find the values of trigonometric functions for angles that are double or half the size of a given angle.
Here is some examples of code implementation in Python:
import math
a = math.pi/3
b = math.pi/4
sin_sum = math.sin(a + b)
sin_diff = math.sin(a  b)
print("sin(a + b) =", sin_sum)
print("sin(a  b) =", sin_diff)
output:
sin(a + b) = 0.9659258262890682
sin(a  b) = 0.3947841760435743
In this example, we have defined the angles a and b to be equal to pi/3 and pi/4, respectively. We then use the math library in Python to calculate the sine of the sum and difference of these angles using the formulas given above.
Another example, in numpy:
import numpy as np
a = np.pi/3
b = np.pi/4
sin_sum = np.sin(a + b)
sin_diff = np.sin(a  b)
print("sin(a + b) =", sin_sum)
print("sin(a  b) =", sin_diff)
output:
“
Sure, the sine double angle formula is just one of many trigonometric identities that are used in mathematics and physics. Here are a few other important identities that are related to the sine double angle formula:

The cosine double angle formula:
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
cos(a – b) = cos(a)cos(b) + sin(a)sin(b) 
The tangent double angle formula:
tan(a + b) = (tan(a) + tan(b))/(1 – tan(a)tan(b))
tan(a – b) = (tan(a) – tan(b))/(1 + tan(a)tan(b)) 
The cotangent double angle formula:
cot(a + b) = (cot(a)cot(b) – 1)/(cot(a) + cot(b))
cot(a – b) = (cot(a)cot(b) + 1)/(cot(a) – cot(b)) 
The Pythagorean identities:
sin^2(x) + cos^2(x) = 1
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = csc^2(x) 
The reciprocal identities:
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)
All these identities can be used to simplify trigonometric expressions, and they are all related to each other in some way. For example, the sine double angle formula can be used to derive the cosine double angle formula, and the Pythagorean identities can be used to derive the reciprocal identities.
In addition to these identities, there are also many other trigonometric identities that are used in mathematics and physics, such as the producttosum identities, the sumtoproduct identities, and the halfangle identities.
It's also worth mentioning that the sine double angle formula and the other identities mentioned above can also be used in the complex plane, in complex numbers representation.
And, in the engineering field, the sine double angle formula is used to calculate the power output of a threephase alternating current system. This is because the power output of a threephase system is the vector sum of the power outputs of the three individual phases. And the sine double angle formula can be used to find the vector sum of these three phase using phasors.
In coding, the abovementioned identities can be easily implemented using libraries such as numpy and math, the same way we have shown in the previous examples.
Popular questions

What is the sine double angle formula?
The sine double angle formula is a fundamental trigonometric identity that states: sin(a + b) = sin(a)cos(b) + cos(a)sin(b) and sin(a – b) = sin(a)cos(b) – cos(a)sin(b) where a and b are the angles being considered. 
How is the sine double angle formula derived?
The sine double angle formula can be derived using the addition formula for sine, which states that: sin(a + b) = sin(a)cos(b) + cos(a)sin(b). It can also be derived by using the complex exponential representation of sine and cosine. 
What are some applications of the sine double angle formula?
The sine double angle formula is used in many areas of mathematics and physics. In physics, it is used to calculate the motion of a simple harmonic oscillator, such as a mass on a spring. In engineering, it is used to calculate the power output of a threephase alternating current system. In trigonometry, it is used to find the values of trigonometric functions for angles that are double or half the size of a given angle. 
How can the sine double angle formula be implemented in code?
The sine double angle formula can be implemented in code using libraries such as math or numpy in Python. For example, the following code calculates the sine of the sum and difference of two angles, a and b:
import math
a = math.pi/3
b = math.pi/4
sin_sum = math.sin(a + b)
sin_diff = math.sin(a  b)
print("sin(a + b) =", sin_sum)
print("sin(a  b) =", sin_diff)
 Are there any related identities to the sine double angle formula?
Yes, there are many related identities to the sine double angle formula. Some examples include the cosine double angle formula, the tangent double angle formula, the cotangent double angle formula, the Pythagorean identities, and the reciprocal identities. All of these identities can be used to simplify trigonometric expressions and are related to each other in some way.
Tag
Trigonometry.