The Correct Answer and Explanation is:
To find the exact value of cos(−195∘)\cos(-195^\circ)cos(−195∘) using a sum or difference formula, we can break the angle into two components whose cosine values are easier to calculate. Let’s use the sum formula for cosine: cos(A+B)=cos(A)cos(B)−sin(A)sin(B)\cos(A + B) = \cos(A)\cos(B) – \sin(A)\sin(B)cos(A+B)=cos(A)cos(B)−sin(A)sin(B)
In this case, we can express −195∘-195^\circ−195∘ as a sum of −180∘-180^\circ−180∘ and −15∘-15^\circ−15∘ because these angles are easier to work with: −195∘=−180∘−15∘-195^\circ = -180^\circ – 15^\circ−195∘=−180∘−15∘
Now, apply the cosine sum formula: cos(−195∘)=cos(−180∘−15∘)=cos(−180∘)cos(−15∘)−sin(−180∘)sin(−15∘)\cos(-195^\circ) = \cos(-180^\circ – 15^\circ) = \cos(-180^\circ)\cos(-15^\circ) – \sin(-180^\circ)\sin(-15^\circ)cos(−195∘)=cos(−180∘−15∘)=cos(−180∘)cos(−15∘)−sin(−180∘)sin(−15∘)
Step 1: Find the trigonometric values for −180∘-180^\circ−180∘ and −15∘-15^\circ−15∘
- cos(−180∘)=−1\cos(-180^\circ) = -1cos(−180∘)=−1 (since cosine is an even function)
- sin(−180∘)=0\sin(-180^\circ) = 0sin(−180∘)=0 (since sine is an odd function)
Now for −15∘-15^\circ−15∘, we know the standard values:
- cos(−15∘)=cos(15∘)\cos(-15^\circ) = \cos(15^\circ)cos(−15∘)=cos(15∘) (since cosine is an even function)
- sin(−15∘)=−sin(15∘)\sin(-15^\circ) = -\sin(15^\circ)sin(−15∘)=−sin(15∘) (since sine is an odd function)
Thus, we need the exact values for cos(15∘)\cos(15^\circ)cos(15∘) and sin(15∘)\sin(15^\circ)sin(15∘), which are:
- cos(15∘)=6+24\cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}cos(15∘)=46+2
- sin(15∘)=6−24\sin(15^\circ) = \frac{\sqrt{6} – \sqrt{2}}{4}sin(15∘)=46−2
Step 2: Substitute into the sum formula
cos(−195∘)=(−1)⋅(6+24)−0⋅(6−24)\cos(-195^\circ) = (-1) \cdot \left( \frac{\sqrt{6} + \sqrt{2}}{4} \right) – 0 \cdot \left( \frac{\sqrt{6} – \sqrt{2}}{4} \right)cos(−195∘)=(−1)⋅(46+2)−0⋅(46−2)
Simplifying: cos(−195∘)=−6+24\cos(-195^\circ) = – \frac{\sqrt{6} + \sqrt{2}}{4}cos(−195∘)=−46+2
Thus, the exact value of cos(−195∘)\cos(-195^\circ)cos(−195∘) is: −6+24\boxed{-\frac{\sqrt{6} + \sqrt{2}}{4}}−46+2
