Problem 64 - Algebra 2

Question

Which expressions are equivalent? 1\. \((\sin \theta)(\csc \theta-\sin \theta)\) II. \(\sin ^{2} \theta-1\) III. \(\cos ^{2} \theta\) A. I and II only B. II and ll only C.l and III only D. \(1,11,\) and \(\| 1\)

Step-by-Step Solution

Answer

The equivalent expressions are I and III only, making the answer 'C' accordingly.

Step 1: Analyze Given Expressions

The given expressions are \((\sin \theta)(\csc \theta-\sin \theta)\), \(\sin ^{2} \theta-1\) and \(\cos ^{2} \theta\).

Step 2: Simplify First Expression

Simplify the first expression: \((\sin \theta)(\csc \theta-\sin \theta) = \sin \theta \cdot \csc \theta - \sin^{2} \theta\). As cscθ is reciprocal of sinθ, this turns into \(1 - \sin^{2} \theta\).

Step 3: Verify Equality with Second Expression

Comparing the simplified version of the first expression, \(1 - \sin^{2} \theta\) with the second expression, it's found they are not equal.

Step 4: Verify Equality with Third Expression

Now comparing \(1 - \sin^{2} \theta\) with the third expression \(\cos ^{2} \theta\). From the Pythagorean identity of trigonometric functions, it's known that \(\cos ^{2} \theta = 1 - \sin^{2} \theta\). Therefore, I and III are equivalent.

Step 5: Verify the Second and Third Expressions

Comparing \(\sin ^{2} \theta-1\) and \(\cos ^{2} \theta\), we can easily see that they are not equal because no identity condition satisfies this equality.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental trigonometric identities derived from the Pythagorean theorem. It establishes a relationship between sine and cosine functions. The identity is given as:

• \(egin{equation} ext{cos}^2 heta + ext{sin}^2 heta = 1 \ ext{or equivalently} \ ext{cos}^2 heta = 1 - ext{sin}^2 heta \ ext{and} \ ext{sin}^2 heta = 1 - ext{cos}^2 heta \ ext{where} \ ext{cos} ( heta) = rac{ ext{adjacent}}{ ext{hypotenuse}} \ ext{and} \ ext{sin} ( heta) = rac{ ext{opposite}}{ ext{hypotenuse}} \\)

This identity is indispensable for simplifying trigonometric expressions.
By understanding its variations, such as \( ext{cos}^2 heta = 1 - ext{sin}^2 heta\), students can transform complex equations into more manageable forms.
It is used extensively to prove and simplify equations in trigonometry.
Identifying this relationship between square functions of sine and cosine helps in converting expressions, like in the given exercise, from one form to another.

Expression simplification

Expression simplification involves reducing trigonometric expressions to a more manageable or recognizable form.
In the context of the given exercise, simplifying the expression \((\sin \theta)(\csc \theta - \sin \theta)\) requires knowing a few basic identities.

• \(\csc \theta\), or cosecant, is the reciprocal of sine, so \(\csc \theta = \frac{1}{\sin \theta}\).

Using this, the original expression becomes:
\( (\sin \theta)(\csc \theta - \sin \theta) = \sin \theta \cdot \frac{1}{\sin \theta} - \sin^2 \theta\), and simplifies to \(1 - \sin^2 \theta\).
Simplification helps to directly compare expressions in their simplest form, making it easier to decide if they are equivalent.
Understanding how to apply basic identities and arithmetic can greatly aid in unraveling what initially seems complex.

Equivalent expressions

Identifying equivalent expressions involves recognizing that two different-looking mathematical statements have the same value.
This concept is crucial in trigonometry, where different algebraic manipulations can reveal equivalence among expressions.
To determine if expressions are equivalent, as demonstrated in the exercise, we simplify each expression using known identities.
For instance, after simplifying \((\sin \theta)(\csc \theta - \sin \theta)\) to \(1 - \sin^2 \theta\), the expression becomes recognizable.
Utilizing the Pythagorean identity, we can identify this as being equivalent to \(\cos^2 \theta\).

• \(1 - \sin^2 \theta = \cos^2 \theta\)

This is a straightforward process if you recall basic identities.
Another expression, \(\sin^2 \theta - 1\), does not simplify into a known identity that matches with the other simplified forms, thus proving they are not equivalent.
Spotting equivalent expressions allows for a better understanding of trigonometric functions and fosters skill in manipulating these identities for other mathematical uses.