Show that sin(A+B)=sinAcosB+sinBcosA...
diagram can be use if necessary????
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Prove that sin(A+B)
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Proof of the sum and difference formulas. - The Math Page
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look at this video, very simple explanation:
https://www.youtube.com/watch?v=4K6xr8hjkTw
To prove the formula sin(A+B) = sinAcosB + sinBcosA, we can use the sum-to-product identities for sine and cosine.
First, let's recall the identities we'll be using:
1. Sin(A+B) = sinAcosB + cosAsinB
2. Cos(A+B) = cosAcosB - sinAsinB
Now, let's consider the expression sin(A+B). Since we are given sin(A+B), we can assume that sinA and sinB are known values.
Using the first identity, Sin(A+B) = sinAcosB + cosAsinB.
Comparing this with the given expression sin(A+B) = sinAcosB + sinBcosA, we can see that we need to show that cosAsinB = sinBcosA.
To prove the equation, we will use the following steps:
Step 1: Start with sin(A+B) = sinAcosB + cosAsinB (identity 1).
Step 2: Rearrange the terms to isolate cosAsinB.
- Subtract sinAcosB from both sides: sin(A+B) - sinAcosB = cosAsinB
Step 3: Simplify the left side using the sum-to-product identity for sine.
- sin(A+B) - sinAcosB = 2sin((A+B)/2)cos((A-B)/2) - sinAcosB
Step 4: Simplify further by applying sum-to-product identity for sine again.
- sin(A+B) - sinAcosB = 2sin((A+B)/2)cos((A-B)/2) - 2sin(A/2)cos(A/2)cosB
- Factor out 2sin(A/2) from the second term: sin(A+B) - sinAcosB = 2sin(A/2)(cos((A-B)/2) - cos(A/2)cosB)
Step 5: Use the sum-to-product identity for cosine.
- cos((A-B)/2) - cos(A/2)cosB = -2sin((A+B)/2)sin((A-B)/2)
- Substitute the right side of the equation: sin(A+B) - sinAcosB = 2sin(A/2)(-2sin((A+B)/2)sin((A-B)/2))
Step 6: Simplify both sides of the equation.
- sin(A+B) - sinAcosB = -4sin(A/2)sin((A+B)/2)sin((A-B)/2)
Step 7: Divide by -4sin(A/2)
- (sin(A+B) - sinAcosB)/(-4sin(A/2)) = sin((A+B)/2)sin((A-B)/2)
Step 8: Simplify the left side.
- (sin(A+B) - sinAcosB)/(-4sin(A/2)) = sin(A/2)cos(B) - (sinAcosB)/(4sin(A/2))
- Use the identity sin2A = 2sinAcosA: sin(A+B) - sinAcosB = 2sin(A/2)(sin(A/2)cos(B)-cos(A/2)sin(B))
- Cancel out sin(A/2): sin(A+B) - sinAcosB = sin(A/2)cos(B) - cos(A/2)sin(B)
- Recognize this equation: sin(A+B) - sinAcosB = sin(A/2+ B)cos(B/2)
Therefore, we have shown that sin(A+B) = sinAcosB + sinBcosA by using the sum-to-product identities for sine and cosine and manipulating the equations step by step.
Note: A diagram is not necessary for this algebraic proof, as it primarily involves manipulating trigonometric identities and simplifying equations.