Find the exact value of cos 68 cos 38 + sin 68 sin 38.
The Correct Answer and Explanation is:
The given expression is:cos(68∘)cos(38∘)+sin(68∘)sin(38∘)\cos(68^\circ)\cos(38^\circ) + \sin(68^\circ)\sin(38^\circ)cos(68∘)cos(38∘)+sin(68∘)sin(38∘)
This is a trigonometric identity. It resembles the sum-to-product identity for cosine: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)=cos(A−B)
Here, A=68∘A = 68^\circA=68∘ and B=38∘B = 38^\circB=38∘. Applying the identity:cos(68∘)cos(38∘)+sin(68∘)sin(38∘)=cos(68∘−38∘)\cos(68^\circ)\cos(38^\circ) + \sin(68^\circ)\sin(38^\circ) = \cos(68^\circ – 38^\circ)cos(68∘)cos(38∘)+sin(68∘)sin(38∘)=cos(68∘−38∘)
Now, simplify the angle:68∘−38∘=30∘68^\circ – 38^\circ = 30^\circ68∘−38∘=30∘
So, the expression simplifies to:cos(30∘)\cos(30^\circ)cos(30∘)
The exact value of cos(30∘)\cos(30^\circ)cos(30∘) is known to be:cos(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}cos(30∘)=23
Therefore, the exact value of the original expression is:32\frac{\sqrt{3}}{2}23
Explanation:
This solution leverages a fundamental trigonometric identity to simplify the given expression. The sum-to-product identity helps convert the sum of cosines and sines into a simpler cosine expression with a difference of angles. Recognizing that cos(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}cos(30∘)=23 allows for a straightforward evaluation. This identity is particularly useful when dealing with sums of trigonometric functions with angles that are easily recognizable.
