Sum/Difference & Double Angle Formulas

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What is the sum formula for tangent?

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The sum formula for tangent is: tan(α + β) = \frac{tan α + tan β}{1 - tan α * tan β}

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What is the difference formula for tangent?

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The difference formula for tangent is: tan(α - β) = \frac{tan α - tan β}{1 + tan α * tan β}

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What is the double angle formula for sine?

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The double angle formula for sine is: sin(2θ) = 2 * sin(θ) * cos(θ)

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What is the double angle formula for cosine?

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The double angle formula for cosine is: cos(2θ) = cos²(θ) - sin²(θ) or cos(2θ) = 2 * cos²(θ) - 1 or cos(2θ) = 1 - 2 * sin²(θ)

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What is the Pythagorean identity?

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The Pythagorean identity states that: sin²(θ) + cos²(θ) = 1

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How do you find sin(105°) using sum formulas?

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To find sin(105°), use the sum formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°) = \frac{\sqrt{3}}{2} * \frac{\sqrt{2}}{2} + \frac{1}{2} * \frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}.

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What is the value of sin(2θ) if cos(θ) = 4/5 and 270° < θ < 360°?

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Using the double angle formula: sin(2θ) = 2 * sin(θ) * cos(θ). First, find sin(θ) using the Pythagorean identity: sin²(θ) = 1 - cos²(θ) = 1 - (4/5)² = 9/25, so sin(θ) = -3/5 (since θ is in the fourth quadrant). Thus, sin(2θ) = 2 * (-3/5) * (4/5) = -24/25.

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What is the value of cos²(x) + sin²(x)?

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The value of cos²(x) + sin²(x) is always equal to 1.

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What is the formula for sin(2u)?

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The formula for sin(2u) is: sin(2u) = 2 * sin(u) * cos(u).

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What is the sum formula for sine?

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The sum formula for sine is: sin(α + β) = sin(α)cos(β) + cos(α)sin(β).