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A vector [tex]\( R \)[/tex] is resolved into its components, [tex]\( R_x \)[/tex] and [tex]\( R_y \)[/tex]. If the ratio [tex]\(\frac{R_x}{R_y}\)[/tex] is 2, what is the angle that the resultant makes with the horizontal?

A. [tex]\( 60.02^{\circ} \)[/tex]
B. [tex]\( 52.60^{\circ} \)[/tex]
C. [tex]\( 26.56^{\circ} \)[/tex]
D. [tex]\( 69.59^{\circ} \)[/tex]
E. [tex]\( 72.35^{\circ} \)[/tex]


Sagot :

To determine the angle that the resultant vector [tex]\( R \)[/tex] makes with the horizontal axis, we can use the given ratio of [tex]\(\frac{R_x}{R_y} = 2\)[/tex].

1. Identify the relationship:
Given [tex]\(\frac{R_x}{R_y} = 2\)[/tex], we can write:
[tex]\[ R_x = 2R_y \][/tex]

2. Use the tangent function:
The angle [tex]\(\theta\)[/tex] that the resultant vector [tex]\(R\)[/tex] makes with the horizontal axis can be found using the tangent function:
[tex]\[ \tan(\theta) = \frac{R_y}{R_x} \][/tex]

3. Substitute the values:
Substitute [tex]\(R_x = 2R_y\)[/tex] into the tangent equation:
[tex]\[ \tan(\theta) = \frac{R_y}{2R_y} = \frac{1}{2} \][/tex]

4. Calculate the angle in radians:
Find the angle [tex]\(\theta\)[/tex] by taking the arctangent of [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \theta = \arctan\left(\frac{1}{2}\right) \][/tex]

5. Convert from radians to degrees:
The arctangent of [tex]\(\frac{1}{2}\)[/tex] in radians can be converted to degrees using the conversion factor [tex]\(180^\circ/\pi\)[/tex]:
[tex]\[ \theta \approx 26.57^\circ \][/tex]

Therefore, the angle that the resultant vector [tex]\(R\)[/tex] makes with the horizontal is approximately:

[tex]\[ \boxed{26.56^\circ} \][/tex]