Simplifying Sin[3θ]: A Guide to Simplifying Trigonometric Expressions

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In summary, the formula for simplifying Sin[3θ] is Sin[3θ] = 3Sinθ - 4Sin^3θ. To solve for θ in Sin[3θ], isolate Sinθ and use the inverse Sin function. Sin[3θ] can be further simplified using the double angle formula, resulting in Sin[3θ] = 6SinθCos^2θ - 4Sin^3θ. Sin[3θ] is used in trigonometry to find values of trigonometric functions and in various identities and equations. Special cases occur when θ is 0, π/2, or π, resulting in a value of 0, 1, or -1
  • #1
Confusingmeh
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Homework Statement


Simplify Sin[3θ] (no double angles)

Homework Equations


Sin[3θ]=Sin[θ+2θ]
Sin[α + β] = Sin[α] Cos[β] + Cos[α] Sin[β]
Sin[3θ] = 3sinθ - 4sin^3 θ
Cos2θ = 1-2Sin^2θ

The Attempt at a Solution


Sin[3θ] = sin[θ+2θ]
Sin[θ+2θ] = Sinθ Cos2θ + Cosθ Sin2θ
Sinθ Cos2θ + Cosθ Sin2θ = Sinθ (1-2Sin^2θ) + Cosθ Sin2θ
=Sinθ - 2Sin^3θ + Cosθ Sin2θ This is where i get lost...I know it should end up being
= (3sinθ - 4sin^3 θ) but i cannot figure out how to get there. I don't know whether I just don't have the formula i need or what.
 
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  • #2
Sin2θ = 2sinθcosθ, remember that one?
 

1. What is the formula for simplifying Sin[3θ]?

The formula for simplifying Sin[3θ] is: Sin[3θ] = 3Sinθ - 4Sin^3θ.

2. How do you solve for θ in Sin[3θ]?

To solve for θ in Sin[3θ], you must first isolate Sinθ by dividing both sides of the equation by 3. Then, use the inverse Sin function to find the value of θ.

3. Can Sin[3θ] be simplified further?

Yes, Sin[3θ] can be further simplified using the double angle formula: Sin[2θ] = 2SinθCosθ. This would result in the final simplified form of Sin[3θ] = 6SinθCos^2θ - 4Sin^3θ.

4. How is Sin[3θ] used in trigonometry?

Sin[3θ] is used in trigonometry to find the value of a trigonometric function at a specific angle θ. It can also be used in various trigonometric identities and equations to simplify and solve for unknown values.

5. Are there any special cases when simplifying Sin[3θ]?

Yes, when θ is a special angle such as 0, π/2, or π, the value of Sin[3θ] will be 0, 1, or -1 respectively. This is due to the periodic nature of the Sin function.

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