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Magazine Chapter 4: Problem 20
Reduce the given expression to a single trigonometric function. $$ \frac{\sin \alpha \tan \alpha}{\operatorname{css} \alpha}+\frac{\sin \alpha}{\operatorname{secc} \alpha} $$
Short Answer
Expert verified
The reduced expression is \(\tan \alpha\).
Step by step solution
01
Recall Trigonometric Identities
In this exercise, we use the trigonometric identities for cosecant and secant. We know that \(\csc \alpha = \frac{1}{\sin \alpha}\) and \(\sec \alpha = \frac{1}{\cos \alpha}\). These identities will help us simplify each term.
02
Rewrite Each Term with Trigonometric Identities
Rewrite the first term: \(\frac{\sin \alpha \tan \alpha}{\csc \alpha}\) as \(\sin \alpha \tan \alpha \cdot \sin \alpha\). Rewrite the second term: \(\frac{\sin \alpha}{\sec \alpha}\) as \(\sin \alpha \cos \alpha\).
03
Simplify the First Term
Simplify \(\sin \alpha \tan \alpha \cdot \sin \alpha\). Since \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\), this becomes \(\frac{\sin^2 \alpha \cdot \sin \alpha}{\cos \alpha} = \frac{\sin^3 \alpha}{\cos \alpha}\).
04
Simplify the Second Term
The second term, \(\sin \alpha \cos \alpha\), is already simplified. It remains \(\sin \alpha \cos \alpha\).
05
Common Denominator
Combine the terms into a single expression. To do that, we need a common denominator. The least common denominator is \(\cos \alpha\). Rewrite the expression as \(\frac{\sin^3 \alpha}{\cos \alpha} + \frac{\sin \alpha \cos \alpha}{\cos \alpha}\).
06
Combine the Fractions
Combine the fractions: \(\frac{\sin^3 \alpha + \sin \alpha \cos^2 \alpha}{\cos \alpha}\).
07
Use Pythagorean Identity
Notice that \(\sin \alpha \cdot \cos^2 \alpha\) can be rewritten using \(\cos^2 \alpha + \sin^2 \alpha = 1\), therefore, \(\cos^2 \alpha = 1 - \sin^2 \alpha\). Substitute this into the numerator to get \(\sin^3 \alpha + \sin \alpha (1 - \sin^2 \alpha)\).
08
Simplify the Numerator
Simplify the numerator: \(\sin^3 \alpha + \sin \alpha - \sin^3 \alpha = \sin \alpha\).
09
Complete the Simplification
This leaves us with the simplified expression \(\frac{\sin \alpha}{\cos \alpha}\), which is \(\tan \alpha\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves rewriting complex expressions using trigonometric identities to make them easier to work with. This process is similar to simplifying algebraic fractions or polynomials. The goal is often to reduce an expression to a singular form, such as a single trigonometric function or a more concise combination of functions.
To simplify, start by:
To simplify, start by:
- Substituting known trigonometric identities to replace complex terms with simpler ones.
- Combining like terms to reduce the number of different trigonometric functions in the expression.
- Finding a common denominator when needed to add or subtract fractions.
Trigonometric Functions
Trigonometric functions like sine (\(\sin\)), cosine (\(cos\)), tangent (\(tan\)), and their reciprocals鈥攃osecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\))鈥攁re fundamental in trigonometry. They relate the angles of a triangle to its side lengths.
Here's a quick refresher on these functions:
Here's a quick refresher on these functions:
- Sine (\(\sin\)): Ratio of the opposite side to the hypotenuse.
- Cosine (\(cos\)): Ratio of the adjacent side to the hypotenuse.
- Tangent (\(tan\)): Ratio of the opposite side to the adjacent side, or equivalently \(\sin \theta / \cos \theta\).
- Cosecant (\(\csc\)): The reciprocal of sine, \(1/\sin \theta\).
- Secant (\(\sec\)): The reciprocal of cosine, \(1/\cos \theta\).
Pythagorean Identity
The Pythagorean Identity is a fundamental identity in trigonometry that relates sine and cosine. It states that for any angle \(\alpha\):
\[\sin^2 \alpha + \cos^2 \alpha = 1\]
It is similar to the Pythagorean theorem in geometry, hence the name. This identity can be used to express one trigonometric function in terms of another. For example, \(\cos^2 \alpha\) can be rewritten as \(1 - \sin^2 \alpha\).
This particular identity is beneficial in simplifying expressions, especially when working with squared terms of sine and cosine. In the given exercise, the Pythagorean Identity was pivotal in simplifying the second half of the numerator, transforming \(\sin \alpha \cos^2 \alpha\) into \(\sin \alpha (1 - \sin^2 \alpha)\). This transformation was crucial in achieving the final reduced expression, showcasing the power of this identity in trigonometric simplification.
\[\sin^2 \alpha + \cos^2 \alpha = 1\]
It is similar to the Pythagorean theorem in geometry, hence the name. This identity can be used to express one trigonometric function in terms of another. For example, \(\cos^2 \alpha\) can be rewritten as \(1 - \sin^2 \alpha\).
This particular identity is beneficial in simplifying expressions, especially when working with squared terms of sine and cosine. In the given exercise, the Pythagorean Identity was pivotal in simplifying the second half of the numerator, transforming \(\sin \alpha \cos^2 \alpha\) into \(\sin \alpha (1 - \sin^2 \alpha)\). This transformation was crucial in achieving the final reduced expression, showcasing the power of this identity in trigonometric simplification.
