- #1
Natasha1
- 493
- 9
Why is sin (x+x) = sinx cosx + cosx sinx ? Simple explanation required please
?rindech said:sin (x+y) = sin x cos y + cos x sin y. Simply place an "x" for the "y" in the formula. Noted: sin (2x) = 2sin x cos x.
VietDao29 said:?
This is not correct.
What if I say that: sin(x + y) = sin(x)sin(y) + cos(x)cos(y) + sin(x)cos(y) + sin(y)cos(x) - 1.
It certainly satisfies: sin(2x) = sin(x + x) = 2sin(x)cos(x). But it's not true, right?
This is known as the sum angle formula for sine. It is derived from the trigonometric identity sin(A+B) = sinA cosB + cosA sinB. Since x+x can be rewritten as 2x, the formula becomes sin(2x) = sinx cosx + cosx sinx.
The sum angle formula allows us to simplify and solve more complex trigonometric equations involving sine. It is also useful in various fields such as engineering, physics, and astronomy where trigonometry is applied.
Yes, there are similar sum angle formulas for cosine (cos(A+B) = cosA cosB - sinA sinB) and tangent (tan(A+B) = (tanA + tanB) / (1 - tanA tanB)). These formulas are also derived from the basic trigonometric identities.
Yes, there is also a double angle formula for sine (sin2x = 2sinx cosx) and a half angle formula for sine (sin(x/2) = ±√((1-cosx)/2)). These can be derived from the sum angle formula and are useful in solving trigonometric equations.
The sum angle formula for sine can be used in various real-world scenarios such as calculating the trajectory of a projectile, determining the forces acting on a structure, or analyzing the motion of a pendulum. It is a fundamental concept in trigonometry and has numerous applications in fields such as engineering, physics, and navigation.