Write each expression in terms of sine and cosine, and then simplify so that no quotients appear in the final expression and all functions are of [tex]$\theta$[/tex] only.
[tex]$\sin ^2(-\theta)-\csc ^2(-\theta)+\cos ^2(-\theta)$[/tex]
Answer (1)
Use trigonometric identities to rewrite the expression in terms of sine and cosine.
Simplify the expression using the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 .
Combine terms to obtain a single fraction.
The simplified expression is s i n 2 ( θ ) − c o s 2 ( θ ) , but this contains a quotient, which contradicts the problem statement. Therefore, the problem cannot be solved as stated.
Explanation
Understanding the Problem We are asked to simplify the expression sin 2 ( − θ ) − csc 2 ( − θ ) + cos 2 ( − θ ) in terms of sine and cosine such that no quotients appear in the final expression.
Simplifying the Expression First, we use the identities sin ( − θ ) = − sin ( θ ) and cos ( − θ ) = cos ( θ ) to rewrite the expression: sin 2 ( − θ ) − csc 2 ( − θ ) + cos 2 ( − θ ) = ( − sin ( θ ) ) 2 − csc 2 ( − θ ) + cos 2 ( θ ) = sin 2 ( θ ) − csc 2 ( − θ ) + cos 2 ( θ ) Next, we use the identity csc ( − θ ) = − csc ( θ ) to rewrite the expression: sin 2 ( θ ) − csc 2 ( − θ ) + cos 2 ( θ ) = sin 2 ( θ ) − ( − csc ( θ ) ) 2 + cos 2 ( θ ) = sin 2 ( θ ) − csc 2 ( θ ) + cos 2 ( θ ) Now, we use the identity csc ( θ ) = s i n ( θ ) 1 to rewrite the expression: sin 2 ( θ ) − csc 2 ( θ ) + cos 2 ( θ ) = sin 2 ( θ ) − sin 2 ( θ ) 1 + cos 2 ( θ ) Using the identity sin 2 ( θ ) + cos 2 ( θ ) = 1 , we have: sin 2 ( θ ) − sin 2 ( θ ) 1 + cos 2 ( θ ) = 1 − sin 2 ( θ ) 1 Combining the terms into a single fraction, we get: 1 − sin 2 ( θ ) 1 = sin 2 ( θ ) sin 2 ( θ ) − 1 Since sin 2 ( θ ) + cos 2 ( θ ) = 1 , we have sin 2 ( θ ) − 1 = − cos 2 ( θ ) . Thus, sin 2 ( θ ) sin 2 ( θ ) − 1 = sin 2 ( θ ) − cos 2 ( θ ) = − ( sin ( θ ) cos ( θ ) ) 2 = − cot 2 ( θ ) Since the problem requires the final expression to be in terms of sine and cosine only and no quotients, we cannot leave the answer as − cot 2 ( θ ) . However, we have already expressed the original expression in terms of sine and cosine, and simplified it to 1 − s i n 2 ( θ ) 1 . To remove the quotient, we can multiply the expression by sin 2 ( θ ) to get sin 2 ( θ ) − 1 , which simplifies to − cos 2 ( θ ) . But this is incorrect because we are not allowed to change the value of the expression. The correct approach is to recognize that we cannot eliminate the quotient and still have the same value. Therefore, the simplified expression is − s i n 2 ( θ ) c o s 2 ( θ ) .
Further Simplification Since the problem requires the final expression to be in terms of sine and cosine only and no quotients, the expression cannot be simplified further than s i n 2 ( θ ) s i n 2 ( θ ) − 1 = s i n 2 ( θ ) − c o s 2 ( θ ) . However, the problem states that no quotients should appear in the final expression. This is not possible, so we must have made a mistake. Let's go back to the original expression: sin 2 ( − θ ) − csc 2 ( − θ ) + cos 2 ( − θ ) . We have shown that this is equal to 1 − s i n 2 ( θ ) 1 . If we want to avoid quotients, we can write csc ( θ ) = s i n ( θ ) 1 , so csc 2 ( θ ) = s i n 2 ( θ ) 1 . Thus, the expression is sin 2 ( − θ ) − csc 2 ( − θ ) + cos 2 ( − θ ) = sin 2 ( θ ) − s i n 2 ( θ ) 1 + cos 2 ( θ ) = 1 − s i n 2 ( θ ) 1 . This is the simplest form in terms of sines and cosines. Since we cannot avoid quotients, there must be an error in the problem statement.
Final Answer The problem asks to write the expression in terms of sine and cosine and simplify so that no quotients appear. We have shown that sin 2 ( − θ ) − csc 2 ( − θ ) + cos 2 ( − θ ) = 1 − s i n 2 ( θ ) 1 . Since we cannot remove the quotient, the problem statement is flawed. However, if we must provide an answer, we can say that the expression is equal to 1 − csc 2 ( θ ) . But this is not in terms of sine and cosine only. The expression s i n 2 ( θ ) − c o s 2 ( θ ) is in terms of sine and cosine, but it has a quotient. Therefore, the problem cannot be solved as stated.
Examples
Trigonometric identities are fundamental in physics, especially when dealing with oscillatory motion, waves, and alternating current circuits. For instance, simplifying trigonometric expressions can help in analyzing the behavior of a pendulum or the resonance in an electrical circuit. By rewriting complex expressions into simpler forms using identities, engineers and physicists can more easily predict and control the behavior of these systems. This ensures efficient designs and accurate predictions in various applications.
