Join the IDNLearn.com community and start finding the answers you need today. Ask your questions and receive comprehensive and trustworthy answers from our experienced community of professionals.
Sagot :
To determine the angle that the line [tex]\( x \cos \alpha + y \sin \alpha = p \)[/tex] makes with the positive [tex]\( x \)[/tex]-axis, we need to carefully consider the standard form of a linear equation and the associated trigonometric relationships.
First, recall that the general form of a line equation in Cartesian coordinates is:
[tex]\[ Ax + By + C = 0 \][/tex]
In our given line equation, we can rewrite it to the standard form:
[tex]\[ x \cos \alpha + y \sin \alpha = p \\ or, x \cos \alpha + y \sin \alpha - p = 0 \][/tex]
Here, [tex]\( A = \cos \alpha \)[/tex] and [tex]\( B = \sin \alpha \)[/tex].
The angle [tex]\( \theta \)[/tex] that a line makes with the positive [tex]\( x \)[/tex]-axis can be found using the relationship:
[tex]\[ \tan \theta = -\frac{A}{B} \][/tex]
For the given line equation:
[tex]\[ \tan \theta = -\frac{\cos \alpha}{\sin \alpha} = -\cot \alpha \][/tex]
The line's slope [tex]\( m \)[/tex] is therefore:
[tex]\[ m = -\cot \alpha \][/tex]
Thus, the angle [tex]\( \theta \)[/tex] can be determined as:
[tex]\[ \theta = \alpha \][/tex]
Therefore, the correct option which indicates the angle this line makes with the positive [tex]\( x \)[/tex]-axis is:
[tex]\[ \boxed{\alpha} \][/tex]
Hence, the correct answer is (6).
First, recall that the general form of a line equation in Cartesian coordinates is:
[tex]\[ Ax + By + C = 0 \][/tex]
In our given line equation, we can rewrite it to the standard form:
[tex]\[ x \cos \alpha + y \sin \alpha = p \\ or, x \cos \alpha + y \sin \alpha - p = 0 \][/tex]
Here, [tex]\( A = \cos \alpha \)[/tex] and [tex]\( B = \sin \alpha \)[/tex].
The angle [tex]\( \theta \)[/tex] that a line makes with the positive [tex]\( x \)[/tex]-axis can be found using the relationship:
[tex]\[ \tan \theta = -\frac{A}{B} \][/tex]
For the given line equation:
[tex]\[ \tan \theta = -\frac{\cos \alpha}{\sin \alpha} = -\cot \alpha \][/tex]
The line's slope [tex]\( m \)[/tex] is therefore:
[tex]\[ m = -\cot \alpha \][/tex]
Thus, the angle [tex]\( \theta \)[/tex] can be determined as:
[tex]\[ \theta = \alpha \][/tex]
Therefore, the correct option which indicates the angle this line makes with the positive [tex]\( x \)[/tex]-axis is:
[tex]\[ \boxed{\alpha} \][/tex]
Hence, the correct answer is (6).
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.