Find the expression as the sine or cosine of an angle, cos 100 Degree cos 50 Degree – sin 100 Degrees in 50 Degree
The correct answer and explanation is :
The given expression is:
[
\cos 100^\circ \cos 50^\circ – \sin 100^\circ \sin 50^\circ
]
Step 1: Recognizing the Identity
We recognize that this expression matches the cosine addition formula:
[
\cos(A + B) = \cos A \cos B – \sin A \sin B
]
Comparing the given expression with the formula, we see that:
- ( A = 100^\circ )
- ( B = 50^\circ )
So, we can rewrite the expression as:
[
\cos(100^\circ + 50^\circ)
]
Step 2: Simplifying
[
\cos(150^\circ)
]
Step 3: Finding the Exact Value
From trigonometric values, we know:
[
\cos 150^\circ = -\cos 30^\circ
]
Since ( \cos 30^\circ = \frac{\sqrt{3}}{2} ), we get:
[
\cos 150^\circ = -\frac{\sqrt{3}}{2}
]
Conclusion:
Thus, the original expression simplifies to:
[
-\frac{\sqrt{3}}{2}
]
Explanation (300 Words)
The given trigonometric expression involves cosine and sine functions, and it closely follows the pattern of the cosine sum identity:
[
\cos(A + B) = \cos A \cos B – \sin A \sin B
]
By recognizing this identity, we substitute ( A = 100^\circ ) and ( B = 50^\circ ), leading us to:
[
\cos(100^\circ + 50^\circ) = \cos 150^\circ
]
Since ( \cos 150^\circ ) is known from standard trigonometric values, we express it in terms of a known angle:
[
\cos 150^\circ = -\cos 30^\circ
]
With ( \cos 30^\circ = \frac{\sqrt{3}}{2} ), this simplifies further to:
[
-\frac{\sqrt{3}}{2}
]
This method is an application of fundamental trigonometric identities, helping to simplify complex expressions efficiently.