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]$\sin ^2(-\theta)+\tan ^2(-\theta)+\cos ^2(-\theta)$[/tex]
Answer (1)
Use negative angle properties to simplify the expression.
Express tan ( θ ) in terms of sin ( θ ) and cos ( θ ) .
Apply the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 .
Recognize that the problem's conditions are contradictory, as it's impossible to eliminate the quotient while keeping the expression in terms of sine and cosine only.
Explanation
Analyzing the Problem Let's analyze the given expression and use trigonometric identities to simplify it. We aim to express everything in terms of sine and cosine and eliminate any quotients.
Using Negative Angle Properties First, we use the properties of sine, cosine, and tangent for negative angles:
sin ( − θ ) = − sin ( θ )
cos ( − θ ) = cos ( θ )
tan ( − θ ) = − tan ( θ )
Substituting these into the expression, we get:
sin 2 ( − θ ) + tan 2 ( − θ ) + cos 2 ( − θ ) = ( − sin ( θ ) ) 2 + ( − tan ( θ ) ) 2 + ( cos ( θ ) ) 2
= sin 2 ( θ ) + tan 2 ( θ ) + cos 2 ( θ )
Expressing Tangent in Terms of Sine and Cosine Next, we express tan ( θ ) in terms of sine and cosine:
tan ( θ ) = cos ( θ ) sin ( θ )
So, the expression becomes:
sin 2 ( θ ) + cos 2 ( θ ) sin 2 ( θ ) + cos 2 ( θ )
Applying the Pythagorean Identity To eliminate the quotient, we want to get rid of the fraction. However, the problem states that no quotients should appear in the final expression, but it doesn't restrict us from using them in intermediate steps. Instead of eliminating the quotient directly, let's try a different approach. We can use the Pythagorean identity sin 2 ( θ ) + cos 2 ( θ ) = 1 :
sin 2 ( θ ) + cos 2 ( θ ) + cos 2 ( θ ) sin 2 ( θ ) = 1 + cos 2 ( θ ) sin 2 ( θ )
Attempting to Eliminate the Quotient Now, we want to eliminate the quotient. To do this, we can rewrite sin 2 ( θ ) as 1 − cos 2 ( θ ) in the fraction:
1 + cos 2 ( θ ) 1 − cos 2 ( θ ) = 1 + cos 2 ( θ ) 1 − cos 2 ( θ ) cos 2 ( θ ) = 1 + cos 2 ( θ ) 1 − 1 = cos 2 ( θ ) 1
This simplifies to c o s 2 ( θ ) 1 , which is sec 2 ( θ ) . However, the problem asks for the final expression to be in terms of sine and cosine only, and without quotients. Let's go back to step 4.
1 + cos 2 ( θ ) sin 2 ( θ )
We need to eliminate the quotient. Notice that if the problem didn't require the final expression to be without quotients, we could have stopped here and written the answer as 1 + tan 2 ( θ ) = sec 2 ( θ ) . However, we need to get rid of the quotient. There must have been a mistake in the problem statement, because it is impossible to get rid of the quotient.
Re-evaluating the Expression and Identifying the Contradiction Let's reconsider the original expression after applying the negative angle identities: sin 2 ( θ ) + cos 2 ( θ ) sin 2 ( θ ) + cos 2 ( θ ) Combining the first and last terms using the Pythagorean identity, we have: 1 + cos 2 ( θ ) sin 2 ( θ ) To remove the quotient, we would need to somehow cancel out the cos 2 ( θ ) in the denominator. However, there's no direct way to do this without introducing more complex terms or violating the condition of having no quotients. It seems the problem's conditions are contradictory, as we cannot eliminate the quotient and keep the expression in terms of sine and cosine only.
Final Answer Given the constraints of the problem, it is impossible to satisfy all conditions simultaneously. The expression simplifies to 1 + c o s 2 ( θ ) s i n 2 ( θ ) , and we cannot eliminate the quotient while keeping the expression in terms of sine and cosine only.
Examples
Understanding trigonometric identities is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, simplifying trigonometric expressions helps in modeling and predicting system behavior. These simplifications allow engineers to design more efficient and reliable systems.
