Problem 54
Question
determine whether each statement makes sense or does not make sense, and explain your reasoning. I expressed sin \(13^{\circ} \cos 48^{\circ}\) as \(\frac{1}{2}\left(\sin 61^{\circ}-\sin 35^{\circ}\right)\)
Step-by-Step Solution
Verified Answer
No, the statement does not make sense. The identity given by the student does not hold true as it is not following any known trigonometric product-to-sum identities. Furthermore, sin and cos functions are not commutative, meaning that sine of a product cannot be represented in the form given by the student.
1Step 1: Understanding the Product-to-Sum formulas
Product-to-Sum Formulas are sin(a)cos(b) = \( \frac{1}{2}[ \sin(a+b) + \sin(a-b) ] \) or -\( \frac{1}{2}[ \sin(a+b) - \sin(a-b) ] \). In this problem, however, we have a sine of a product which is not directly usable with our product-to-sum formulas. This is a clue that the student's transformation was incorrect.
2Step 2: Trying the transformation
Following the student's logic, the expression sin(13° cos 48°) supposedly equals \( \frac{1}{2}( \sin 61^{\circ} - \sin 35^{\circ}) \). Let's try and verify this by checking if the identities hold true for our product-to-sum formulas. Upon comparing, we see that the given expression doesn't match the Product-to-Sum Formulas, therefore, the conversion done by the student seems to be incorrect since the expression used is not in the format of these formulae.
3Step 3: Validating the statement
If we calculate both sides separately using a calculator, we can validate the statement. Given that sin and cos functions are not commutative, the expression sin(13° cos 48°) cannot be represented as \( \frac{1}{2}( \sin 61^{\circ} - \sin 35^{\circ}) \). Hence, the statement does not make sense.
Key Concepts
Product-to-Sum FormulasSine and Cosine FunctionsMathematical Reasoning
Product-to-Sum Formulas
The product-to-sum formulas are important tools in trigonometry. They help in transforming products of sine and cosine functions into sums or differences. This transformation can simplify the computation or analysis of trigonometric expressions. It involves these specific formulas:
However, the key detail is the expression involves a multiplication inside the sine function—\( \sin(13^\circ \cos 48^\circ) \). This form does not match the product-to-sum templates, as those explicitly start with separate sine and cosine functions multiplying together. The confusion arises because the expression is not in the same format required to apply these formulas, indicating a misapplication of the identity.
- \( \sin(a)\cos(b) = \frac{1}{2}[\sin(a+b) + \sin(a-b)] \)
- \( \cos(a)\sin(b) = \frac{1}{2}[\sin(a+b) - \sin(a-b)] \)
However, the key detail is the expression involves a multiplication inside the sine function—\( \sin(13^\circ \cos 48^\circ) \). This form does not match the product-to-sum templates, as those explicitly start with separate sine and cosine functions multiplying together. The confusion arises because the expression is not in the same format required to apply these formulas, indicating a misapplication of the identity.
Sine and Cosine Functions
The sine and cosine functions are fundamental in trigonometry. They are periodic and represent the coordinates on a unit circle. Specifically:
This repetition makes them invaluable in modeling wave-like or cyclic phenomena.
In the context of this exercise, understanding how these functions behave helps to clarify why the initial transformation of \( \sin(13^\circ \cos 48^\circ) \) doesn't fit the explained identities. Sine and cosine are separate entities within these formulas, and their direct multiplication is covered, not embedding the cosine inside the sine as happened in the student's statement.
- \( \sin(\theta) \) gives the y-coordinate of a point on the unit circle at an angle \( \theta \)
- \( \cos(\theta) \) gives the x-coordinate
This repetition makes them invaluable in modeling wave-like or cyclic phenomena.
In the context of this exercise, understanding how these functions behave helps to clarify why the initial transformation of \( \sin(13^\circ \cos 48^\circ) \) doesn't fit the explained identities. Sine and cosine are separate entities within these formulas, and their direct multiplication is covered, not embedding the cosine inside the sine as happened in the student's statement.
Mathematical Reasoning
Mathematical reasoning is the logical thought process of determining the correctness or incorrectness of statements using mathematical principles. It involves examining assumptions, logic, and consistency.
This exercise illustrates the importance of careful mathematical reasoning in understanding and applying trigonometric identities.
To validate expressions using identities, it is crucial to correctly identify the form and ensure it matches the predefined identity's format. In this example, the student's attempt to express \( \sin(13^\circ \cos 48^\circ) \) as \( \frac{1}{2}( \sin 61^\circ - \sin 35^\circ) \) fails because it doesn't logically follow from the product-to-sum identity.
By comparing both sides and calculating values if necessary, one applies mathematical reasoning to confirm the initial misalignment and understand why such a transformation doesn't hold true.
This exercise illustrates the importance of careful mathematical reasoning in understanding and applying trigonometric identities.
To validate expressions using identities, it is crucial to correctly identify the form and ensure it matches the predefined identity's format. In this example, the student's attempt to express \( \sin(13^\circ \cos 48^\circ) \) as \( \frac{1}{2}( \sin 61^\circ - \sin 35^\circ) \) fails because it doesn't logically follow from the product-to-sum identity.
By comparing both sides and calculating values if necessary, one applies mathematical reasoning to confirm the initial misalignment and understand why such a transformation doesn't hold true.
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