Problem 57
Question
Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\sin \theta(\csc \theta-\sin \theta)$$
Step-by-Step Solution
Verified Answer
The simplified expression is \( \cos^2 \theta \).
1Step 1: Substitute Reciprocal Identity
The expression given is \( \sin \theta(\csc \theta - \sin \theta) \). The reciprocal identity states that \( \csc \theta = \frac{1}{\sin \theta} \). Substitute this into the expression: \( \sin \theta \left( \frac{1}{\sin \theta} - \sin \theta \right) \).
2Step 2: Simplify the Expression
Inside the parentheses, simplify \( \frac{1}{\sin \theta} - \sin \theta \). This becomes \( \frac{1 - \sin^2 \theta}{\sin \theta} \).
3Step 3: Recognize the Pythagorean Identity
Recall the Pythagorean identity: \( 1 - \sin^2 \theta = \cos^2 \theta \). Substitute this into the expression: \( \frac{\cos^2 \theta}{\sin \theta} \).
4Step 4: Distribute Sine Term
The expression \( \sin \theta \cdot \frac{\cos^2 \theta}{\sin \theta} \) simplifies to just \( \cos^2 \theta \) after canceling \( \sin \theta \) in the numerator and denominator.
Key Concepts
Reciprocal IdentityPythagorean IdentityTrigonometric Simplification
Reciprocal Identity
The reciprocal identity is a fundamental concept in trigonometry used to simplify expressions. It states that the cosecant of an angle, denoted as \( \csc \theta \), is the reciprocal of the sine of that angle. In mathematical terms, this is expressed as \( \csc \theta = \frac{1}{\sin \theta} \). This identity is often useful when working to convert trigonometric expressions into simpler forms.
Understanding reciprocal identities not only helps in simplifying the given expressions but also in solving more advanced trigonometric problems. It allows students to transform more complex formulas into expressions that are easier to handle. In the exercise presented, using the reciprocal identity allowed us to replace \( \csc \theta \) and start simplifying the expression.
Understanding reciprocal identities not only helps in simplifying the given expressions but also in solving more advanced trigonometric problems. It allows students to transform more complex formulas into expressions that are easier to handle. In the exercise presented, using the reciprocal identity allowed us to replace \( \csc \theta \) and start simplifying the expression.
Pythagorean Identity
Another critical concept is the Pythagorean identity in trigonometry. This identity is a statement about the squares of the trigonometric functions sine and cosine. It says that for any angle \( \theta \): \( \sin^2 \theta + \cos^2 \theta = 1 \).
In practical uses, especially when simplifying trigonometric expressions, this identity is often rearranged or manipulated to express one function in terms of the other. For instance, as seen in the exercise, it allows us to rewrite \( 1 - \sin^2 \theta \) as \( \cos^2 \theta \).
This step is instrumental in further simplifying expressions. By understanding and applying the Pythagorean identity, students can see how expressions interrelate and streamline them effectively, moving from a complex to a straightforward form.
In practical uses, especially when simplifying trigonometric expressions, this identity is often rearranged or manipulated to express one function in terms of the other. For instance, as seen in the exercise, it allows us to rewrite \( 1 - \sin^2 \theta \) as \( \cos^2 \theta \).
This step is instrumental in further simplifying expressions. By understanding and applying the Pythagorean identity, students can see how expressions interrelate and streamline them effectively, moving from a complex to a straightforward form.
Trigonometric Simplification
Trigonometric simplification is a process that involves using various identities to rewrite a trig expression so that it is easier to understand or solve. This process often involves replacing trigonometric functions with equivalents to condense expressions or solve equations.
In the provided exercise, simplification was achieved by recognizing and applying the reciprocal and Pythagorean identities. These identities helped convert the given expression from a more complicated form \( \sin \theta(\csc \theta - \sin \theta) \) into \( \cos^2 \theta \), a simpler and more workable expression.
Effective simplification helps in reducing errors, especially in solving trigonometric equations in mathematics and applications in physics and engineering. Mastering these small yet potent steps can significantly enhance problem-solving skills across STEM fields.
In the provided exercise, simplification was achieved by recognizing and applying the reciprocal and Pythagorean identities. These identities helped convert the given expression from a more complicated form \( \sin \theta(\csc \theta - \sin \theta) \) into \( \cos^2 \theta \), a simpler and more workable expression.
Effective simplification helps in reducing errors, especially in solving trigonometric equations in mathematics and applications in physics and engineering. Mastering these small yet potent steps can significantly enhance problem-solving skills across STEM fields.
Other exercises in this chapter
Problem 57
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Use a calculator to give each real-number value of \(y .\) $$y=\arccos 0.44624593$$
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