\frac{\sin \left(180^{\circ}- B \right) \cdot \tan \left(360^{\circ}- B \right)}{\sin \left(180^{\circ}+ B \right)}
Answer (1)
Use the trigonometric identity sin ( 18 0 ∘ − x ) = sin ( x ) .
Use the trigonometric identity tan ( 36 0 ∘ − x ) = − tan ( x ) .
Use the trigonometric identity sin ( 18 0 ∘ + x ) = − sin ( x ) .
Simplify the expression to get the final answer: tan ( B ) .
Explanation
Understanding the Problem We are asked to simplify the given trigonometric expression: sin ( 18 0 ∘ + B ) sin ( 18 0 ∘ − B ) ⋅ tan ( 36 0 ∘ − B )
Listing Trigonometric Identities We will use the following trigonometric identities to simplify the expression:
sin ( 18 0 ∘ − x ) = sin ( x )
tan ( 36 0 ∘ − x ) = − tan ( x )
sin ( 18 0 ∘ + x ) = − sin ( x )
Applying the Identities Applying these identities to the given expression, we get: sin ( 18 0 ∘ + B ) sin ( 18 0 ∘ − B ) ⋅ tan ( 36 0 ∘ − B ) = − sin ( B ) sin ( B ) ⋅ ( − tan ( B ))
Simplifying the Expression Now, we simplify the expression by cancelling out the common terms. We can cancel sin ( B ) from the numerator and the denominator: − sin ( B ) sin ( B ) ⋅ ( − tan ( B )) = − 1 − tan ( B ) = tan ( B )
Final Answer Therefore, the simplified expression is tan ( B ) .
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and navigation. For instance, when analyzing the motion of a pendulum or the trajectory of a projectile, simplifying trigonometric expressions helps in determining angles, distances, and velocities. In navigation, these identities are used to calculate bearings and distances, ensuring accurate positioning and course plotting. Moreover, in signal processing, trigonometric functions and their identities are essential for analyzing and manipulating signals, such as audio or radio waves.
