Find x. Round your answer to the nearest tenth of a degree

Find x. Round your answer to the nearest tenth of a degree.

The Correct Answer and Explanation is :

To solve for ( x ) in the triangle provided, we can use the Law of Sines, which relates the sides of a triangle to the sines of its angles. The Law of Sines states that for any triangle with sides ( a ), ( b ), and ( c ) and angles ( A ), ( B ), and ( C ) opposite those sides, respectively, the following relationship holds:

[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
]

From the diagram, we have the following information:

( A = 40^\circ ) (angle at the top)
( b = 23 ) (side opposite ( A ))
( c = 25 ) (the hypotenuse of the triangle)
( B = x ) (angle opposite side ( b ))

We are trying to find angle ( B ) (or ( x )). Using the Law of Sines:

[
\frac{23}{\sin 40^\circ} = \frac{25}{\sin x}
]

We can solve for ( \sin x ) as follows:

[
\sin x = \frac{25 \cdot \sin 40^\circ}{23}
]

First, let’s calculate ( \sin 40^\circ ) and then solve for ( x ).

import math

# Constants
sin_40 = math.sin(math.radians(40))
ratio = 25 * sin_40 / 23

# Solve for x
x_degrees = math.degrees(math.asin(ratio))
x_degrees

After obtaining the value of ( x ), I’ll provide a detailed explanation of the calculation.

The angle ( x ) is approximately ( 44.3^\circ ).

Explanation:

To find ( x ), we used the Law of Sines which states that the ratio of a side of a triangle to the sine of its opposite angle is the same for all three sides and their corresponding angles. Here’s the step-by-step breakdown of the calculation:

Using the Law of Sines: We know from the triangle that:
[
\frac{23}{\sin 40^\circ} = \frac{25}{\sin x}
]
This sets up our equation based on the known angle and its opposite side, and the unknown angle ( x ) and its opposite side.
Rearranging the Equation: To isolate ( \sin x ), we rearrange the formula:
[
\sin x = \frac{25 \cdot \sin 40^\circ}{23}
]
Calculating ( \sin 40^\circ ): We first calculate ( \sin 40^\circ ) using a calculator or Python. The sine of an angle in a right triangle represents the ratio of the length of the opposite side to the hypotenuse.
Substituting Values: Plugging the value of ( \sin 40^\circ ) into our equation gives us the sine of angle ( x ).
Finding ( x ): Finally, using the arcsine function (inverse of sine), we calculate ( x ) from ( \sin x ). This function returns the angle whose sine is the given number, providing us the measure of angle ( x ) in degrees.

This process allows us to accurately determine the unknown angle in a non-right triangle when we have one angle and two sides. By applying these trigonometric principles, we can solve a wide range of problems involving triangles in various fields such as engineering, physics, and navigation.

Explanation:

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