\frac{\sin ^2\left(360^{\circ}-\theta\right) \cdot \cos ^2\left(360^{\circ}-\theta\right)}{\sin \left(180^{\circ}+\theta\right) \cdot \sin \left(180^{\circ}-\theta\right)}
Answer (1)
The given trigonometric expression simplifies as follows:
Apply trigonometric identities to rewrite the expression.
Simplify the numerator and the denominator.
Cancel out the common sin 2 ( θ ) terms.
The simplified expression is − cos 2 ( θ ) .
− cos 2 ( θ )
Explanation
Understanding the Problem We are asked to simplify the trigonometric expression:
Expression to Simplify sin ( 18 0 ∘ + θ ) ⋅ sin ( 18 0 ∘ − θ ) sin 2 ( 36 0 ∘ − θ ) ⋅ cos 2 ( 36 0 ∘ − θ )
Trigonometric Identities We will use the following trigonometric identities to simplify the expression:
sin ( 36 0 ∘ − θ ) = − sin ( θ )
cos ( 36 0 ∘ − θ ) = cos ( θ )
sin ( 18 0 ∘ + θ ) = − sin ( θ )
sin ( 18 0 ∘ − θ ) = sin ( θ )
Substitution Substituting these identities into the expression, we get:
( − sin ( θ )) ⋅ ( sin ( θ )) ( − sin ( θ ) ) 2 ⋅ ( cos ( θ ) ) 2
Simplifying the expression:
Simplification − sin 2 ( θ ) sin 2 ( θ ) ⋅ cos 2 ( θ )
Final Simplification Canceling out the common terms, we obtain:
− cos 2 ( θ )
Therefore, the simplified expression is − cos 2 ( θ ) .
Examples
Understanding trigonometric identities is crucial in various fields such as physics, engineering, and computer graphics. For example, when analyzing the motion of a pendulum or designing acoustic systems, simplifying trigonometric expressions helps in modeling and predicting system behavior. In computer graphics, these identities are used to manipulate and render images, ensuring accurate and efficient visual representation.
