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Magazine Chapter 6: Problem 86
$$\text { Simplify } \frac{\cos ^{3}(x)+\cos (x) \sin ^{2}(x)}{\sin (x)}$$
Short Answer
Expert verified
\( \cot(x) \)
Step by step solution
01
Factor out common terms in the numerator
Notice that the numerator \(\frac{\text{cos}^{3}(x) + \text{cos}(x) \text{sin}^{2}(x)}{\text{sin}(x)}\) has a common factor of \(\text{cos}(x)\). Factor out \(\text{cos}(x)\).
02
Rewrite the numerator
Rewrite the numerator as \(\text{cos}(x)(\text{cos}^{2}(x) + \text{sin}^{2}(x))\). Using the Pythagorean identity \(\text{cos}^{2}(x) + \text{sin}^{2}(x) = 1\), simplify the expression inside the parentheses.
03
Simplify the expression inside the parentheses
Since \(\text{cos}^{2}(x) + \text{sin}^{2}(x) = 1\), the expression simplifies to \(\text{cos}(x) \times 1 \). Thus, the numerator becomes \(\text{cos}(x)\).
04
Write the simplified fraction
Now write the fraction as \(\frac{\text{cos}(x)}{\text{sin}(x)}\).
05
Use the cotangent identity
Recall that \(\frac{\text{cos}(x)}{\text{sin}(x)} = \text{cot}(x)\). Therefore, the simplified form of the original expression is \(\text{cot}(x)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
cosine
The cosine function, often abbreviated as \(\text{cos}\), is one of the primary trigonometric functions. It relates the adjacent side of a right-angle triangle to its hypotenuse. For an angle \(x\) in a right triangle, the cosine is given by:
\[ \text{cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Here are some key points to remember:
\[ \text{cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Here are some key points to remember:
- The cosine function is periodic, with a period of \(2\pi\), meaning \(\text{cos}(x + 2\pi) = \text{cos}(x)\).
- It is an even function, so \(\text{cos}(-x) = \text{cos}(x)\).
- The cosine function ranges from -1 to 1.
sine
The sine function, denoted as \(\text{sin}\), is another fundamental trigonometric function. It relates the opposite side of a right-angle triangle to its hypotenuse. For an angle \(x\), the sine is defined by:
\[ \text{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Important attributes of the sine function include:
\[ \text{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Important attributes of the sine function include:
- It is periodic with a period of \(2\pi\), which implies \(\text{sin}(x + 2\pi) = \text{sin}(x)\).
- It is an odd function, resulting in \(\text{sin}(-x) = -\text{sin}(x)\).
- The range of sine is between -1 and 1.
Pythagorean identity
The Pythagorean identity is a critical equation in trigonometry. It states:
\[ \text{cos}^{2}(x) + \text{sin}^{2}(x) = 1 \]
This identity is derived from the Pythagorean Theorem and is true for any angle \(x\). Some applications include:
\[ \text{cos}^{2}(x) + \text{sin}^{2}(x) = 1 \]
This identity is derived from the Pythagorean Theorem and is true for any angle \(x\). Some applications include:
- Simplifying trigonometric expressions, such as in our original problem.
- Converting between \(\text{cos}(x)\) and \(\text{sin}(x)\).
- Checking the correctness of other trigonometric identities and proofs.
cotangent
The cotangent function, symbolized as \(\text{cot}\), is the reciprocal of the tangent function. It is defined for an angle \(x\) by:
\[ \text{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)} \]
Key characteristics of \(\text{cot}(x)\) include:
\[ \text{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)} \]
Key characteristics of \(\text{cot}(x)\) include:
- It is periodic with a period of \(\pi\), so \(\text{cot}(x + \pi) = \text{cot}(x)\).
- The function is undefined when \(\text{sin}(x) = 0\), e.g., at \(x = k\pi\) where \(k\) is an integer.
- It varies from \(-\infty\) to \(\infty\).
