Give four angles in radians where y =tan(x) has vertical asymptotes.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Transcribed Image Text:**Question:** Give four angles in radians where \( y = \tan(x) \) has vertical asymptotes.
**Explanation:**
In the function \( y = \tan(x) \), vertical asymptotes occur at the points where the function is undefined. This happens when the cosine of \( x \) (i.e., \( \cos(x) \)) is equal to zero because tangent is defined as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
**Key Points for Educational Context:**
- The \( \tan(x) \) function has vertical asymptotes at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is any integer.
- Therefore, four such angles in radians where vertical asymptotes occur can be given by starting with some specific values of \( k \).
**Examples:**
1. \( x = \frac{\pi}{2} \) (where \( k = 0 \))
2. \( x = \frac{3\pi}{2} \) (where \( k = 1 \))
3. \( x = -\frac{\pi}{2} \) (where \( k = -1 \))
4. \( x = \frac{5\pi}{2} \) (where \( k = 2 \))
These angles represent the points along the x-axis at which the tangent function will have vertical asymptotes.
![## Trigonometric Identities
Trigonometric identities are fundamental tools in mathematics, especially in the fields of geometry, calculus, and engineering. Here are some key trigonometric identities that you might find useful:
1. **Double Angle Identities:**
\[
\sin(2u) = 2 \sin(u) \cos(u)
\]
\[
\cos(2u) = \cos^2(u) - \sin^2(u)
\]
2. **Half Angle Identities:**
\[
\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
\[
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
### Explanation:
- **Double Angle Identities**:
- The **sine of double an angle** identity shows that sine of twice any angle \(2u\) can be expressed as \(2\sin(u)\cos(u)\).
- The **cosine of double an angle** identity expresses the cosine of twice an angle \(2u\) as the difference of the square of cosine (\(\cos^2(u)\)) and square of sine (\(\sin^2(u)\)) of the angle \(u\).
- **Half Angle Identities**:
- The **sine of half an angle** identity states that the sine of half an angle \(\theta/2\) is equal to the positive or negative square root of \(\frac{1 - \cos(\theta)}{2}\). The sign depends on the quadrant in which the angle \(\theta/2\) lies.
- The **cosine of half an angle** identity states that the cosine of half an angle \(\theta/2\) is equal to the positive or negative square root of \(\frac{1 + \cos(\theta)}{2}\). Again, the sign depends on the quadrant in which the angle \(\theta/2\) lies.
Understanding and applying these identities can help simplify complex trigonometric problems and are particularly useful in solving integrals, derivatives, and equations involving trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d6261b1-d641-4c15-8b49-7d4d38863372%2F333b39ec-422d-4556-af5c-a3ac24277ff5%2Fm6fh7_processed.png&w=3840&q=75)
Transcribed Image Text:## Trigonometric Identities
Trigonometric identities are fundamental tools in mathematics, especially in the fields of geometry, calculus, and engineering. Here are some key trigonometric identities that you might find useful:
1. **Double Angle Identities:**
\[
\sin(2u) = 2 \sin(u) \cos(u)
\]
\[
\cos(2u) = \cos^2(u) - \sin^2(u)
\]
2. **Half Angle Identities:**
\[
\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
\[
\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
\]
### Explanation:
- **Double Angle Identities**:
- The **sine of double an angle** identity shows that sine of twice any angle \(2u\) can be expressed as \(2\sin(u)\cos(u)\).
- The **cosine of double an angle** identity expresses the cosine of twice an angle \(2u\) as the difference of the square of cosine (\(\cos^2(u)\)) and square of sine (\(\sin^2(u)\)) of the angle \(u\).
- **Half Angle Identities**:
- The **sine of half an angle** identity states that the sine of half an angle \(\theta/2\) is equal to the positive or negative square root of \(\frac{1 - \cos(\theta)}{2}\). The sign depends on the quadrant in which the angle \(\theta/2\) lies.
- The **cosine of half an angle** identity states that the cosine of half an angle \(\theta/2\) is equal to the positive or negative square root of \(\frac{1 + \cos(\theta)}{2}\). Again, the sign depends on the quadrant in which the angle \(\theta/2\) lies.
Understanding and applying these identities can help simplify complex trigonometric problems and are particularly useful in solving integrals, derivatives, and equations involving trigonometric functions.
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