Problem 59 - Algebra 2

Question

Which expressions are equivalent? \(I .-\tan \left(\frac{\pi}{2}-\theta\right)\) \(\quad\) Il. \(\cos (-\theta)\) \(\quad\) III. \(\tan \left(-\left(\frac{\pi}{2}-\theta\right)\right)\) F. I and II only \(\quad\) G. II and lll only \(\quad\) H. l and \(Il\) only \(\quad\) J. \(I,II,\) and III

Step-by-Step Solution

Answer

III and I are equivalent, so the correct answer choice is H.

Step 1: Analyzing Expression I

The expression \(I .-\tan \left(\frac{\pi}{2}-\theta\right)\) can be simplified using the identity \(\tan(90^\circ - x) = \cot(x)\). Hence, the expression I becomes \(-\cot(\theta)\)

Step 2: Analyzing Expression II

The expression \(\cos (-\theta)\) works with the property that the cosine function is even, which means \(\cos(-\theta) = \cos(\theta)\). Hence, expression II remains as \(\cos(\theta)\)

Step 3: Analyzing Expression III

The expression \(\tan \left(-\left(\frac{\pi}{2}-\theta\right)\right)\) can be simplified using the property that the tangent function is odd, which means \(\tan(-x) = -\tan(x)\). Therefore, this expression becomes \(-\tan \left(\frac{\pi}{2}-\theta\right)\). Applying the identity \(\tan(90^\circ - x) = \cot(x)\) like in Expression I, the expression III becomes \(-\cot(\theta)\)

Step 4: Comparing the Expressions

Expression I and III are equivalent since both are equal to \(-\cot(\theta)\). Expression II is not equivalent to the other two as it simplifies to \(\cos(\theta)\)

Key Concepts

Even and odd functions

In mathematics, functions are categorized into two key types: even and odd functions. This categorization is based on how these functions behave when their inputs are negated.

• Even Functions: These functions satisfy the condition \( f(-x) = f(x) \) for every value of \( x \) in their domain. A classic example is the cosine function, \( \cos(x) \). This symmetry about the y-axis makes even functions unique.
• Odd Functions: For odd functions, the condition is \( f(-x) = -f(x) \). Examples include the sine and tangent functions, such as \( \tan(x) \). They have rotational symmetry about the origin.

Understanding these properties helps simplify expressions. In the original problem, we used the even property of cosine and the odd property of tangent to simplify the trigonometric expressions.

Angle subtraction identity

The angle subtraction identity is a vital trigonometric formula that simplifies expressions involving angles. Specifically, the identity for tangent of an angle subtraction is given by \( \tan(90^\circ - x) = \cot(x) \). This identity helps convert tangent expressions involving a subtraction of angles into a more manageable form.
For instance, in expression I, \( -\tan\left(\frac{\pi}{2} - \theta\right) \) simplifies to \( -\cot(\theta) \) using this identity. This conversion can be extremely useful when trying to identify equivalences between different trigonometric expressions.
Mastering angle subtraction identities supports quick recognition of trigonometric forms and minimizes calculation errors, especially in more complicated trigonometric manipulations.

Trigonometric equivalence

Trigonometric equivalence revolves around recognizing expressions that fundamentally represent the same values. It involves reshaping trigonometric expressions using various identities and properties.
In the given exercise, we determined which expressions were equivalent by transforming each using known identities. Expression I and III both resulted in \( -\cot(\theta) \) thanks to the angle subtraction identity and the odd property of tangent.

• Expression I: Starts as \( -\tan\left(\frac{\pi}{2} - \theta\right) \), simplifies to \( -\cot(\theta) \).
• Expression III: Begins as \( \tan\left(-\left(\frac{\pi}{2}-\theta\right)\right) \), simplifies similarly to \( -\cot(\theta) \).
• Expression II: \( \cos(-\theta) = \cos(\theta) \) remains as a standalone result due to its even nature.

Recognizing equivalent expressions offers deeper insights into the structural relationships between trigonometric functions and aids in solving complex trigonometric problems efficiently.