Use identities to fill in the blank with the appropriate trigonometric function name.
[tex]$\csc 66^{\circ}=\frac{1}{24^{\circ}}$[/tex]
[tex]$\csc 66^{\circ}=\frac{1}{\square 24^{\circ}}$[/tex]
Answer (1)
Recognize the cofunction identity csc ( θ ) = sec ( 9 0 ∘ − θ ) .
Apply the identity to transform csc 6 6 ∘ into sec ( 9 0 ∘ − 6 6 ∘ ) .
Simplify the expression to find csc 6 6 ∘ = sec 2 4 ∘ .
Conclude that the missing trigonometric function is secant: sec .
Explanation
Problem Analysis We are given the equation csc 6 6 ∘ = □ 2 4 ∘ 1 and we need to find the appropriate trigonometric function to fill in the blank.
Cofunction Identity Recall the cofunction identities. A relevant identity is csc ( θ ) = sec ( 9 0 ∘ − θ ) .
Applying the Identity Apply this identity to csc 6 6 ∘ : csc 6 6 ∘ = sec ( 9 0 ∘ − 6 6 ∘ ) = sec ( 2 4 ∘ ) .
Finding the Missing Function Therefore, the missing trigonometric function is secant. We can write the equation as csc 6 6 ∘ = sec 2 4 ∘ .
Verifying the Answer Since sec ( x ) = c o s ( x ) 1 , we can rewrite the equation as csc 6 6 ∘ = c o s 2 4 ∘ 1 . However, the question asks for a trigonometric function directly, not its reciprocal. The correct identity to use is csc 6 6 ∘ = sec 2 4 ∘ , and since sec x = c o s x 1 , we are looking for the function that is the reciprocal of cosine, which is secant.
Final Answer Thus, the equation is csc 6 6 ∘ = sec 2 4 ∘ .
Examples
Understanding trigonometric identities like cofunction relationships is crucial in fields such as physics and engineering. For instance, when analyzing wave behavior or designing electrical circuits, these identities allow engineers to simplify complex equations and make accurate predictions. Knowing that csc ( x ) = sec ( 9 0 ∘ − x ) helps in converting between different trigonometric forms, which can be essential for optimizing system performance and ensuring accurate calculations in real-world applications.
