$\frac{\cos \left(360^{\circ}-\theta\right) \cdot \tan \left(180^{\circ}-\theta\right)}{\cos \left(180^{\circ}-\theta\right)}$
Answer (1)
∙ Use the trigonometric identities cos ( 36 0 ∘ − θ ) = cos ( θ ) , tan ( 18 0 ∘ − θ ) = − tan ( θ ) , and cos ( 18 0 ∘ − θ ) = − cos ( θ ) .
∙ Substitute the identities into the expression: − c o s ( θ ) c o s ( θ ) ⋅ ( − t a n ( θ )) .
∙ Cancel out the common factor cos ( θ ) .
∙ Simplify the expression to get the final answer: tan ( θ ) .
Explanation
Understanding the Problem We are asked to simplify the trigonometric expression: cos ( 18 0 ∘ − θ ) cos ( 36 0 ∘ − θ ) ⋅ tan ( 18 0 ∘ − θ )
Recalling Trigonometric Identities We will use trigonometric identities to simplify the expression. Recall the following identities:
cos ( 36 0 ∘ − θ ) = cos ( − θ ) = cos ( θ )
tan ( 18 0 ∘ − θ ) = − tan ( θ )
cos ( 18 0 ∘ − θ ) = − cos ( θ )
Substituting the Identities Substitute these identities into the original expression: cos ( 18 0 ∘ − θ ) cos ( 36 0 ∘ − θ ) ⋅ tan ( 18 0 ∘ − θ ) = − cos ( θ ) cos ( θ ) ⋅ ( − tan ( θ ))
Simplifying the Expression Now, simplify the expression by canceling out the common factors. We can cancel out cos ( θ ) from the numerator and the denominator: − cos ( θ ) cos ( θ ) ⋅ ( − tan ( θ )) = − 1 − tan ( θ ) = tan ( θ )
Final Answer Therefore, the simplified expression is tan ( θ ) .
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and computer graphics. For example, when analyzing the motion of a pendulum, simplifying expressions involving angles is crucial for determining the pendulum's period and amplitude. Similarly, in electrical engineering, simplifying trigonometric expressions helps in analyzing alternating current (AC) circuits and understanding the relationships between voltage, current, and phase angles. In computer graphics, these identities are used to perform rotations and transformations of objects in 3D space, ensuring that the objects appear correctly on the screen.
