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Chapter 7: Problem 53

Simplify. Check your results using a graphing calculator. \(\cos (\pi-x)+\cot x \sin \left(x-\frac{\pi}{2}\right)\)

Short Answer

Expert verified
-\text{cos}x - \frac{\text{cos}^2x}{\text{sin}x}

Step by step solution

01

Simplify \(\text{cos}(\pi - x)\)

Using the trigonometric identity \(\text{cos}(a - b) = \text{cos}a \text{cos}b + \text{sin}a \text{sin}b\), we can rewrite \(\text{cos}(\pi - x)\) as: \[ \text{cos}(\pi - x) = -\text{cos}(x) \]
02

Simplify \(\text{sin}\left(x-\frac{\pi}{2}\right)\)

Using the trigonometric identity \(\text{sin}(a - b) = \text{sin}a \text{cos}b - \text{cos}a \text{sin}b\), we can rewrite \(\text{sin}\left(x-\frac{\pi}{2}\right)\) as: \[ \text{sin}\left(x-\frac{\pi}{2}\right) = \text{sin}x\text{cos}\left(\frac{\pi}{2}\right) - \text{cos}x\text{sin}\left(\frac{\pi}{2}\right) \] Since \(\text{cos}\left(\frac{\pi}{2}\right) = 0\) and \(\text{sin}\left(\frac{\pi}{2}\right) = 1\), this simplifies to: \[ \text{sin}\left(x-\frac{\pi}{2}\right) = -\text{cos}x \]
03

Simplify \(\text{cot}x \text{sin}\left(x-\frac{\pi}{2}\right)\)

We know that \(\text{cot}x = \frac{\text{cos}x}{\text{sin}x}\). Using the result from Step 2, we have: \[ \text{cot}x \text{sin}\left(x-\frac{\pi}{2}\right) = \frac{\text{cos}x}{\text{sin}x} \cdot (-\text{cos}x) = -\frac{\text{cos}^2x}{\text{sin}x} \]
04

Combine the results

Add the simplified terms from Step 1 and Step 3: \[ \text{cos}(\pi - x) + \text{cot}x \text{sin}\left(x-\frac{\pi}{2}\right) = -\text{cos}x - \frac{\text{cos}^2x}{\text{sin}x} \]
05

Verify with a graphing calculator

To confirm, input \(\text{cos}(\pi - x) + \text{cot}x \text{sin}\left(x-\frac{\pi}{2}\right)\) and \(-\text{cos}x - \frac{\text{cos}^2x}{\text{sin}x}\) into the graphing calculator. Both expressions should produce the same graph, verifying the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cosine identity
The cosine identity \(\text{cos}(a - b) = \text{cos}(a)\text{cos}(b) + \text{sin}(a)\text{sin}(b)\) is a fundamental trigonometric identity often used in simplification. Let's illustrate this with our example in the exercise. When we have \(\text{cos}(\text{蟺} - x)\), we can set \(a = \text{蟺}\) and \(b = x\). Substituting these in, we get: \(\text{cos}(\text{蟺} - x) = \text{cos}(\text{蟺})\text{cos}(x) + \text{sin}(\text{蟺})\text{sin}(x)\).
Since \(\text{cos}(\text{蟺}) = -1\) and \(\text{sin}(\text{蟺}) = 0\), this simplifies to:
\( \text{cos}(\text{蟺} - x) = -\text{cos}(x) \).
By understanding and applying the cosine identity correctly, we simplify trigonometric expressions effectively, making complex problems more manageable.
sine identity
The sine identity \(\text{sin}(a - b) = \text{sin}(a)\text{cos}(b) - \text{cos}(a)\text{sin}(b)\) helps in breaking down sine terms into more straightforward components. Using our exercise as a reference, we need to simplify \(\text{sin}(x - \frac{\text{蟺}}{2})\).
Here, let \(a = x\) and \(b = \frac{\text{蟺}}{2}\). Applying the identity, we get: \(\text{sin}(x - \frac{\text{蟺}}{2}) = \text{sin}(x)\text{cos}(\frac{\text{蟺}}{2}) - \text{cos}(x)\text{sin}(\frac{\text{蟺}}{2})\).
Knowing that \(\text{cos}(\frac{\text{蟺}}{2}) = 0\) and \(\text{sin}(\frac{\text{蟺}}{2}) = 1\), it simplifies to:
\(\text{sin}(x - \frac{\text{蟺}}{2}) = -\text{cos}(x)\).
This application of the sine identity offers a clearer path to simplifying expressions involving angles shifted by \(\frac{\text{蟺}}{2}\).
cotangent
The cotangent function, denoted as \(\text{cot}(x)\), is the reciprocal of the tangent function. It is defined as \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\).
In our exercise, we encounter \(\text{cot}(x) \text{sin}(x - \frac{\text{蟺}}{2})\).
Let's break it down: \(\text{cot}(x)\) can be written as \(\frac{\text{cos}(x)}{\text{sin}(x)}\). Using the fact that \(\text{sin}(x - \frac{\text{蟺}}{2}) = -\text{cos}(x)\), we can substitute and simplify as follows: \(\text{cot}(x) \text{sin}(x - \frac{\text{蟺}}{2}) = \frac{\text{cos}(x)}{\text{sin}(x)} \times -\text{cos}(x) = -\frac{\text{cos}^2(x)}{\text{sin}(x)}\).
Understanding the relationship and application of cotangent is crucial in simplifying trigonometric expressions further.

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Most popular questions from this chapter

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a period of \(\pi ?\)

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=3 \sec \theta\) in \(\sqrt{x^{2}-9}\). Then find \(\sin \theta\) and \(\cos \theta\)

Use the sum and difference identities to evaluate exactly. Then check using a graphing calculator. $$\sin \frac{7 \pi}{12}$$

SOLVE. $$\sin ^{-1} x=\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{2}$$

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