Determine the value of the following expression:
$\frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(x-90^{\circ}\right)}{4 \sin \left(360^{\circ}+x\right)}$
Answer (1)
Use trigonometric identities to simplify the expression.
Substitute the identities into the original expression.
Simplify the expression by canceling out common terms.
The final value of the expression is 4 1 .
Explanation
Understanding the Problem We are asked to find the value of the expression 4 sin ( 36 0 ∘ + x ) tan ( 18 0 ∘ − x ) ⋅ sin ( x − 9 0 ∘ ) To do this, we will use trigonometric identities to simplify the expression.
Listing Trigonometric Identities We will use the following trigonometric identities:
tan ( 18 0 ∘ − x ) = − tan ( x )
sin ( x − 9 0 ∘ ) = − cos ( x )
sin ( 36 0 ∘ + x ) = sin ( x )
Substituting the Identities Substitute these identities into the expression: 4 sin ( x ) − tan ( x ) ⋅ − cos ( x ) = 4 sin ( x ) tan ( x ) cos ( x )
Rewriting the Expression Use the identity tan ( x ) = c o s ( x ) s i n ( x ) to rewrite the expression: 4 sin ( x ) c o s ( x ) s i n ( x ) cos ( x )
Simplifying Simplify the expression: 4 sin ( x ) sin ( x )
Final Simplification Cancel out sin ( x ) from the numerator and denominator, assuming sin ( x ) = 0 :
4 1
Examples
Imagine you're designing a robotic arm that needs to operate at various angles. Simplifying trigonometric expressions like this helps engineers predict the arm's position and orientation accurately. By using trigonometric identities, they can optimize the arm's movements, ensuring it reaches the desired locations efficiently. This is crucial for tasks like assembly line work, surgery, or even space exploration, where precision is key.
