Problem 61 - Algebra 2

Question

Which expression is NOT equivalent to \(\cos \theta ?\) \(\begin{array}{llll}{\text { F. }-\sin \left(\theta-90^{\circ}\right)} & {\text { G. }-\cos (-\theta)} & {\text { H. } \sin \left(\theta+90^{\circ}\right)} & {\text { J. }-\cos \left(\theta+180^{\circ}\right)}\end{array}\)

Step-by-Step Solution

Answer

All the expressions, F, G, H, J are equivalent to \(\cos \theta\). Therefore, none of the given options is NOT equivalent to \(\cos \theta\).

Step 1: Analyzing the first expression

Analyze the first expression, -\(\sin \left(\theta-90^{\circ}\right)\). Using the co-function identity, which is \(\cos \theta = \sin \left(90^{\circ} - \theta\right)\), it can be rewritten as \(\cos \left(90^{\circ} + \theta\right)\). So, this expression is equivalent to \(\cos \theta\).

Step 2: Analyzing the second expression

The next thing is to examine is \(-\cos (-\theta)\). Since cosine function is an even function, \(\cos (-\theta) = \cos \theta\). Hence, \(-\cos (-\theta) = -\cos \theta\). Therefore, this expression is also equivalent to \(\cos \theta\).

Step 3: Analyzing the third expression

Now we proceed to examine \(\sin \left(\theta+90^{\circ}\right)\). Using the same co-function identity, this can be rewritten as -\(\cos (\theta)\). So, this expression is also equivalent to \(\cos \theta\).

Step 4: Analyzing the fourth expression

Finally, analyzing \(-\cos \left(\theta+180^{\circ}\right)\), using the periodic property of cosine function \(\cos (theta+180) = - cos (theta)\). Therefore, this becomes --\(\cos (\theta)\) which is \(\cos \theta\). So, this option is also equivalent to \(\cos \theta\).

Key Concepts

Co-function Identities

Co-function identities are useful in relating different trigonometric functions to each other. One of the core identities is \[ \cos\theta = \sin(90^\circ - \theta) \]This identity implies that the cosine of an angle is equal to the sine of its complement. Understanding these relationships can simplify complex trigonometric expressions.
In exercises such as the one discussed:

• Expressions like \sin (\theta + 90^\circ) can be converted using co-function identities.
• It often involves recognizing that the angle in the sine function is related to a cosine expression through the identity.

Using these identities, you can efficiently verify if certain complex expressions are equivalent by rewriting or rearranging terms to show that they match known trigonometric identities.

Even and Odd Functions

Understanding the nature of trigonometric functions as even or odd is crucial in manipulating and simplifying expressions. Cosine is an even function, meaning:\[ \cos (-\theta) = \cos(\theta) \]This means that the cosine function produces the same value for \(-\theta\) as it does for \(\theta\).
In the step-by-step solution mentioned:

• An expression like -\cos(-\theta) utilizes the even property of cosine.
• This property allows for simplification by removing the negative sign from the angle inside the cosine function.

Odd functions such as the sine function, where \( \sin(-\theta) = -\sin(\theta) \), behave differently. Recognizing these properties allows you to determine the equivalence of various trigonometric expressions efficiently.

Periodic Properties of Trigonometric Functions

Trigonometric functions have specific periodic properties which make them repeat their values in predictable patterns. The cosine function is periodic with a period of 360 degrees, meaning:\[ \cos(\theta + 360^\circ) = \cos(\theta) \]However, it is also important to understand half-cycle periodicities such as:\[ \cos(\theta + 180^\circ) = -\cos(\theta) \]This is crucial for solving trigonometric equations and expressions, as seen in the provided problem.

• The periodic nature of the cosine function can simplify an expression like -\cos(\theta + 180^\circ) to be \cos(\theta).
• Recognizing these periodic properties allows you to handle phase shifts and still predict the outcome of trigonometric equations.

Understanding these periodic traits is instrumental for simplifying expressions and matching them with trigonometric identities.