Questions and Answers
What is the definition of sin(2x)?
What is one definition of cos(2x)?
What is the definition of tan(2x)?
2tanx / (1 - tan²x)
What is the definition of sin(x/2)?
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What is the definition of cos(x/2)?
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What is the definition of tan(x/2)?
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Study Notes
Half Angle and Double Angle Identities
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sin(2x) is expressed as 2sinxcosx, representing the sine of double angles using the sine and cosine of the original angle.
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cos(2x) can be represented by various forms:
- cos²x - sin²x, demonstrating the relationship between squares of cosine and sine.
- 2cos²x - 1, allowing transformation using cosine.
- 1 - 2sin²x, providing an alternative expression based on sine.
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tan(2x) is calculated using the formula 2tanx / (1 - tan²x), offering a tangent double angle expression derived from the tangent of the original angle.
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For the half-angle identities:
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sin(x/2) can be expressed as ±√(1-cosx)/2, relating the sine of halved angles to the cosine.
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cos(x/2) is defined by ±√(1+cosx)/2, connecting the cosine of halved angles back to the original cosine value.
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tan(x/2) is determined by ±√(1-cosx)/(1+cosx), showing the tangent of half angles as a function of the cosine.
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These identities are crucial in trigonometry for simplifying expressions and solving equations.
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