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Math  /  Trigonometry

Questionsin⁡(π3−π4)−2sin⁡(4π12−3π12)→\sin \left(\frac{\pi}{3}-\frac{\pi}{4}\right)-2 \sin \left(\frac{4 \pi}{12}-\frac{3 \pi}{12}\right) \rightarrow
Simplify and then give an exact value for each expression. a) cos⁡25∘cos⁡5∘−sin⁡25∘sin⁡5∘\cos 25^{\circ} \cos 5^{\circ}-\sin 25^{\circ} \sin 5^{\circ} cos⁡25∘cos⁡5∘−sin⁡25∘sin⁡5∘⇒cos⁡(25+5)−8cos⁡(30∘)→32\begin{array}{l} \cos 25^{\circ} \cos 5^{\circ}-\sin 25^{\circ} \sin 5^{\circ} \\ \Rightarrow \cos (25+5)-8 \cos \left(30^{\circ}\right) \rightarrow \frac{\sqrt{3}}{2} \end{array} d) cos⁡7π12cos⁡π3+sin⁡7π12sin⁡π3\cos \frac{7 \pi}{12} \cos \frac{\pi}{3}+\sin \frac{7 \pi}{12} \sin \frac{\pi}{3}

Studdy Solution

STEP 1

1. We are simplifying trigonometric expressions using angle sum and difference identities.
2. We will use known exact values for trigonometric functions at specific angles.
3. The problem involves multiple parts, each requiring separate simplification.

STEP 2

1. Simplify the expression sin⁡(π3−π4)−2sin⁡(4π12−3π12)\sin \left(\frac{\pi}{3}-\frac{\pi}{4}\right)-2 \sin \left(\frac{4 \pi}{12}-\frac{3 \pi}{12}\right).
2. Simplify and find the exact value for expression (a).
3. Simplify and find the exact value for expression (d).

STEP 3

Simplify sin⁡(π3−π4)\sin \left(\frac{\pi}{3}-\frac{\pi}{4}\right) using the sine difference identity:
sin⁡(π3−π4)=sin⁡π3cos⁡π4−cos⁡π3sin⁡π4\sin \left(\frac{\pi}{3}-\frac{\pi}{4}\right) = \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}
Substitute the known values:
=(32)(22)−(12)(22)=64−24=6−24= \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}

STEP 4

Simplify 2sin⁡(4π12−3π12)2 \sin \left(\frac{4 \pi}{12}-\frac{3 \pi}{12}\right):
2sin⁡(π3−π4)=2(6−24)=6−222 \sin \left(\frac{\pi}{3} - \frac{\pi}{4}\right) = 2 \left(\frac{\sqrt{6} - \sqrt{2}}{4}\right) = \frac{\sqrt{6} - \sqrt{2}}{2}

STEP 5

Combine the results:
6−24−6−22=6−24−2(6−2)4=−6−24\frac{\sqrt{6} - \sqrt{2}}{4} - \frac{\sqrt{6} - \sqrt{2}}{2} = \frac{\sqrt{6} - \sqrt{2}}{4} - \frac{2(\sqrt{6} - \sqrt{2})}{4} = -\frac{\sqrt{6} - \sqrt{2}}{4}
Simplified result:
−6−24-\frac{\sqrt{6} - \sqrt{2}}{4}

STEP 6

Simplify expression (a) using the cosine difference identity:
cos⁡25∘cos⁡5∘−sin⁡25∘sin⁡5∘=cos⁡(25∘+5∘)=cos⁡30∘\cos 25^{\circ} \cos 5^{\circ} - \sin 25^{\circ} \sin 5^{\circ} = \cos(25^{\circ} + 5^{\circ}) = \cos 30^{\circ}
Exact value:
cos⁡30∘=32\cos 30^{\circ} = \frac{\sqrt{3}}{2}

STEP 7

Simplify expression (d) using the cosine sum identity:
cos⁡7π12cos⁡π3+sin⁡7π12sin⁡π3=cos⁡(7π12−π3)\cos \frac{7 \pi}{12} \cos \frac{\pi}{3} + \sin \frac{7 \pi}{12} \sin \frac{\pi}{3} = \cos\left(\frac{7 \pi}{12} - \frac{\pi}{3}\right)
Calculate the angle:
7π12−π3=7π12−4π12=3π12=π4\frac{7 \pi}{12} - \frac{\pi}{3} = \frac{7 \pi}{12} - \frac{4 \pi}{12} = \frac{3 \pi}{12} = \frac{\pi}{4}
Exact value:
cos⁡π4=22\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}

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