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 given triangle, 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:

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

From the image, we know the following values:

Side ( a = 27 )
Side ( c = 15 )
Angle ( A = 56^\circ )

We need to find angle ( C ), which is denoted as ( x ) in the diagram.

Using the law of sines:

[
\frac{27}{\sin 56^\circ} = \frac{15}{\sin x}
]

First, calculate ( \sin 56^\circ ):

[
\sin 56^\circ \approx 0.8290
]

Plug this value into the equation:

[
\frac{27}{0.8290} = \frac{15}{\sin x}
]

Solve for ( \sin x ):

[
\sin x = \frac{15 \times 0.8290}{27}
]

[
\sin x \approx 0.4611
]

Now, find ( x ) by taking the arcsin:

[
x = \sin^{-1}(0.4611)
]

Let’s calculate the value of ( x ) and then round it to the nearest tenth of a degree.

import math

# Calculate sin x
sin_x = (15 * 0.8290) / 27

# Calculate x
x = math.degrees(math.asin(sin_x))

# Round x to the nearest tenth
x_rounded = round(x, 1)
x_rounded

The value of ( x ) is approximately ( 27.4^\circ ).

Explanation

We used the law of sines to find ( x ) in the given triangle. Here’s a step-by-step breakdown:

Identify Known Values: From the triangle, we have sides ( a = 27 ), ( c = 15 ), and angle ( A = 56^\circ ).
Apply the Law of Sines: We utilized the relationship (\frac{a}{\sin A} = \frac{c}{\sin C}), where ( C ) is the angle we need to find.
Calculate (\sin A): Using ( \sin 56^\circ \approx 0.8290 ).
Rearrange to Find ( \sin C ): The equation becomes (\sin C = \frac{15 \times 0.8290}{27}) which simplifies to (\sin C \approx 0.4611).
Solve for ( C ): Using the inverse sine function, we found ( C = \sin^{-1}(0.4611) ), yielding ( C \approx 27.4^\circ ).

This process accurately determines the angle ( x ) using trigonometric functions and the properties of the triangle given, providing a precise and useful method in the context of geometry and trigonometry.

Explanation

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