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Math  /  Trigonometry

QuestionUse a sum-to-product formula to find the exact value. Write your answer as a simplified fraction and rationalize the denominator, if nece sin⁡165∘−sin⁡75∘=\sin 165^{\circ}-\sin 75^{\circ}= □\square  延 \sqrt{\text { 延 }}

Studdy Solution

STEP 1

1. We are asked to find the exact value of the expression using a sum-to-product formula.
2. The sum-to-product identities can be used to simplify expressions involving sine and cosine.
3. The expression involves angles in degrees.

STEP 2

1. Identify and apply the appropriate sum-to-product formula.
2. Simplify the resulting expression.
3. Rationalize the denominator if necessary.

STEP 3

Identify the sum-to-product formula for sine differences:
The formula is: sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2) \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)
Apply this formula to the given expression sin⁡165∘−sin⁡75∘\sin 165^{\circ} - \sin 75^{\circ}.
A=165∘,B=75∘ A = 165^{\circ}, \quad B = 75^{\circ}
sin⁡165∘−sin⁡75∘=2cos⁡(165∘+75∘2)sin⁡(165∘−75∘2) \sin 165^{\circ} - \sin 75^{\circ} = 2 \cos \left( \frac{165^{\circ} + 75^{\circ}}{2} \right) \sin \left( \frac{165^{\circ} - 75^{\circ}}{2} \right)

STEP 4

Calculate the angles inside the cosine and sine functions:
165∘+75∘2=240∘2=120∘ \frac{165^{\circ} + 75^{\circ}}{2} = \frac{240^{\circ}}{2} = 120^{\circ}
165∘−75∘2=90∘2=45∘ \frac{165^{\circ} - 75^{\circ}}{2} = \frac{90^{\circ}}{2} = 45^{\circ}
Substitute these values back into the expression:
2cos⁡(120∘)sin⁡(45∘) 2 \cos(120^{\circ}) \sin(45^{\circ})

STEP 5

Use known values for trigonometric functions:
cos⁡(120∘)=−12 \cos(120^{\circ}) = -\frac{1}{2} sin⁡(45∘)=22 \sin(45^{\circ}) = \frac{\sqrt{2}}{2}
Substitute these into the expression:
2(−12)(22) 2 \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right)
Simplify the expression:
=−22⋅22=−22 = -\frac{2}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}

STEP 6

The expression is already simplified as a fraction and the denominator is rationalized.
The exact value is:
−22 \boxed{-\frac{\sqrt{2}}{2}}

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