Answer
The solutions for \( \tan(A) \) are:
- \( A = \tan^{-1}\left(\frac{7}{9}\right) + n\pi \)
- \( A = \tan^{-1}\left(-\frac{7}{9}\right) + n\pi \)
- \( A = \tan^{-1}\left(\frac{\sqrt{7}}{9}\right) + n\pi \)
- \( A = \tan^{-1}\left(-\frac{\sqrt{7}}{9}\right) + n\pi \)
where \( n \) is any integer.
Solution
To solve the identity \( \tan(A) = \pm \frac{7}{9} \) and \( \tan(A) = \pm \frac{\sqrt{7}}{9} \), we will analyze each case separately.
### Case 1: \( \tan(A) = \pm \frac{7}{9} \)
1. **Understanding the Tangent Function**: The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. Therefore, if \( \tan(A) = \frac{7}{9} \), we can visualize a right triangle where the opposite side is 7 and the adjacent side is 9.
2. **Finding the Angle**: To find the angle \( A \), we can use the arctangent function:
\[
A = \tan^{-1}\left(\frac{7}{9}\right)
\]
This will give us one solution for \( A \). Since tangent is periodic with a period of \( \pi \), the general solutions will be:
\[
A = \tan^{-1}\left(\frac{7}{9}\right) + n\pi \quad \text{for } n \in \mathbb{Z}
\]
Similarly, for \( \tan(A) = -\frac{7}{9} \):
\[
A = \tan^{-1}\left(-\frac{7}{9}\right) + n\pi \quad \text{for } n \in \mathbb{Z}
\]
### Case 2: \( \tan(A) = \pm \frac{\sqrt{7}}{9} \)
1. **Understanding the Tangent Function**: Similarly, if \( \tan(A) = \frac{\sqrt{7}}{9} \), we can visualize a right triangle where the opposite side is \( \sqrt{7} \) and the adjacent side is 9.
2. **Finding the Angle**: To find the angle \( A \), we can use the arctangent function:
\[
A = \tan^{-1}\left(\frac{\sqrt{7}}{9}\right)
\]
The general solutions will be:
\[
A = \tan^{-1}\left(\frac{\sqrt{7}}{9}\right) + n\pi \quad \text{for } n \in \mathbb{Z}
\]
For \( \tan(A) = -\frac{\sqrt{7}}{9} \):
\[
A = \tan^{-1}\left(-\frac{\sqrt{7}}{9}\right) + n\pi \quad \text{for } n \in \mathbb{Z}
\]
### Summary of Solutions
- For \( \tan(A) = \frac{7}{9} \):
\[
A = \tan^{-1}\left(\frac{7}{9}\right) + n\pi
\]
- For \( \tan(A) = -\frac{7}{9} \):
\[
A = \tan^{-1}\left(-\frac{7}{9}\right) + n\pi
\]
- For \( \tan(A) = \frac{\sqrt{7}}{9} \):
\[
A = \tan^{-1}\left(\frac{\sqrt{7}}{9}\right) + n\pi
\]
- For \( \tan(A) = -\frac{\sqrt{7}}{9} \):
\[
A = \tan^{-1}\left(-\frac{\sqrt{7}}{9}\right) + n\pi
\]
These equations provide the angles \( A \) for each case, where \( n \) is any integer.
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain
Simplify this solution
