Fill in the blank to complete the fundamental identity.
[tex]$\sec (-\theta)=$\square[/tex]
Answer (1)
Use the definition of secant: sec ( − θ ) = c o s ( − θ ) 1 .
Apply the even function property of cosine: cos ( − θ ) = cos ( θ ) .
Substitute to find sec ( − θ ) = c o s ( θ ) 1 .
Use the definition of secant again to get the final answer: sec ( − θ ) = sec ( θ ) .
The final answer is sec ( θ ) .
Explanation
Understanding the Problem We are asked to find the expression that is equal to sec ( − θ ) .
Using the Definition of Secant We know that sec ( x ) = c o s ( x ) 1 . Therefore, sec ( − θ ) = c o s ( − θ ) 1 .
Using the Property of Even Functions Since cosine is an even function, we have cos ( − θ ) = cos ( θ ) .
Substitution Substituting this into the expression for sec ( − θ ) , we get sec ( − θ ) = c o s ( θ ) 1 .
Final Answer Using the definition of secant again, we have c o s ( θ ) 1 = sec ( θ ) . Therefore, sec ( − θ ) = sec ( θ ) .
Examples
Understanding trigonometric identities like sec ( − θ ) = sec ( θ ) is crucial in fields like physics and engineering. For instance, when analyzing alternating current (AC) circuits, the voltage and current waveforms are often modeled using trigonometric functions. The symmetry properties of these functions, such as the even nature of cosine (and thus secant), help simplify calculations and predict the behavior of the circuit under different conditions. This ensures accurate design and efficient operation of electrical systems.
