Sum and Difference Formulas Trigonometry

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Define the Sum Formula for Sine.

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The Sum Formula for Sine states that for any angles \( \alpha \) and \( \beta \): \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \).

Tags

CCSS.HSF.TF.C.9

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Define the Difference Formula for Sine.

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The Difference Formula for Sine states that for any angles \( \alpha \) and \( \beta \): \( \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \).

Tags

CCSS.HSF.TF.C.9

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Define the Sum Formula for Cosine.

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The Sum Formula for Cosine states that for any angles \( \alpha \) and \( \beta \): \( \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta \).

Tags

CCSS.HSF.TF.C.9

Front

Define the Difference Formula for Cosine.

Back

The Difference Formula for Cosine states that for any angles \( \alpha \) and \( \beta \): \( \cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta \).

Tags

CCSS.HSF.TF.C.9

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What is the value of \( \sin(90^{\circ}) \)?

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The value of \( \sin(90^{\circ}) \) is 1.

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CCSS.HSF.TF.A.2

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What is the value of \( \cos(0^{\circ}) \)?

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The value of \( \cos(0^{\circ}) \) is 1.

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CCSS.HSF.TF.A.2

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What is the value of \( \sin(0^{\circ}) \)?

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The value of \( \sin(0^{\circ}) \) is 0.

Tags

CCSS.HSF.TF.A.2

Front

What is the value of \( \cos(90^{\circ}) \)?

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The value of \( \cos(90^{\circ}) \) is 0.

Tags

CCSS.HSF.TF.A.2

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Solve for \( \cos(105^{\circ}) \) using the difference formula.

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Using the difference formula: \( \cos(105^{\circ}) = \cos(90^{\circ} + 15^{\circ}) = \cos(90^{\circ})\cos(15^{\circ}) - \sin(90^{\circ})\sin(15^{\circ}) = 0 \cdot \cos(15^{\circ}) - 1 \cdot \sin(15^{\circ}) = -\sin(15^{\circ}) \).

Tags

CCSS.HSF.TF.C.9

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Calculate \( \sin(\frac{5\pi}{12}) \cos(\frac{\pi}{4}) - \cos(\frac{5\pi}{12}) \sin(\frac{\pi}{4}) \).

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Using the sine difference formula: \( \sin(\frac{5\pi}{12}) \cos(\frac{\pi}{4}) - \cos(\frac{5\pi}{12}) \sin(\frac{\pi}{4}) = \sin(\frac{5\pi}{12} - \frac{\pi}{4}) = \sin(\frac{5\pi}{12} - \frac{3\pi}{12}) = \sin(\frac{2\pi}{12}) = \sin(\frac{\pi}{6}) = \frac{1}{2} \).

Tags

CCSS.HSF.TF.C.9