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Math  /  Trigonometry

QuestionQuestion 13
Use Identities to find the exact value. cos⁡(19π12)\cos \left(\frac{19 \pi}{12}\right) (A) 6−24\frac{\sqrt{6}-\sqrt{2}}{4} (B) 2−6\sqrt{2}-\sqrt{6} (C) 2−64\frac{\sqrt{2}-\sqrt{6}}{4} (D) −6−2-\sqrt{6}-\sqrt{2}

Studdy Solution

STEP 1

1. We need to find the exact value of cos⁡(19π12)\cos \left(\frac{19 \pi}{12}\right).
2. We will use angle sum or difference identities to express 19π12\frac{19 \pi}{12} in terms of known angles.
3. We will use trigonometric identities to simplify the expression.

STEP 2

1. Express 19π12\frac{19 \pi}{12} as a sum or difference of angles with known cosine values.
2. Apply the cosine angle sum or difference identity.
3. Simplify the expression using known trigonometric values.
4. Match the simplified expression to one of the given options.

STEP 3

Express 19π12\frac{19 \pi}{12} in terms of known angles. We can write:
19π12=π+π12\frac{19 \pi}{12} = \pi + \frac{\pi}{12}

STEP 4

Apply the cosine angle sum identity:
cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin b
Here, a=π a = \pi and b=π12 b = \frac{\pi}{12} .
cos⁡(19π12)=cos⁡(π+π12)=cos⁡πcos⁡π12−sin⁡πsin⁡π12\cos \left(\frac{19 \pi}{12}\right) = \cos \left(\pi + \frac{\pi}{12}\right) = \cos \pi \cos \frac{\pi}{12} - \sin \pi \sin \frac{\pi}{12}

STEP 5

Substitute known trigonometric values:
cos⁡π=−1,sin⁡π=0\cos \pi = -1, \quad \sin \pi = 0
Thus,
cos⁡(19π12)=(−1)⋅cos⁡π12−0⋅sin⁡π12\cos \left(\frac{19 \pi}{12}\right) = (-1) \cdot \cos \frac{\pi}{12} - 0 \cdot \sin \frac{\pi}{12}
cos⁡(19π12)=−cos⁡π12\cos \left(\frac{19 \pi}{12}\right) = -\cos \frac{\pi}{12}

STEP 6

Now, find cos⁡π12\cos \frac{\pi}{12} using the half-angle identity:
cos⁡π12=cos⁡(π6−π4)\cos \frac{\pi}{12} = \cos \left(\frac{\pi}{6} - \frac{\pi}{4}\right)
Apply the cosine difference identity:
cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos(a - b) = \cos a \cos b + \sin a \sin b
Here, a=π6 a = \frac{\pi}{6} and b=π4 b = \frac{\pi}{4} .
cos⁡π12=cos⁡π6cos⁡π4+sin⁡π6sin⁡π4\cos \frac{\pi}{12} = \cos \frac{\pi}{6} \cos \frac{\pi}{4} + \sin \frac{\pi}{6} \sin \frac{\pi}{4}

STEP 7

Substitute known trigonometric values:
cos⁡π6=32,sin⁡π6=12,cos⁡π4=22,sin⁡π4=22\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2}, \quad \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}
cos⁡π12=(32)(22)+(12)(22)\cos \frac{\pi}{12} = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
cos⁡π12=64+24\cos \frac{\pi}{12} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
cos⁡π12=6+24\cos \frac{\pi}{12} = \frac{\sqrt{6} + \sqrt{2}}{4}

STEP 8

Substitute back to find cos⁡(19π12)\cos \left(\frac{19 \pi}{12}\right):
cos⁡(19π12)=−cos⁡π12=−(6+24)\cos \left(\frac{19 \pi}{12}\right) = -\cos \frac{\pi}{12} = -\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)
cos⁡(19π12)=−6−24\cos \left(\frac{19 \pi}{12}\right) = \frac{-\sqrt{6} - \sqrt{2}}{4}
This matches option (A).
The exact value of cos⁡(19π12)\cos \left(\frac{19 \pi}{12}\right) is:
6−24\boxed{\frac{\sqrt{6}-\sqrt{2}}{4}}

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