The problem presents a right triangle with one angle of 48 degrees, a hypotenuse of 10, and an unknown side 'y'. We need to find the value of 'y' rounded to the nearest hundredth. To solve this, we need to utilize trigonometric functions. ...
Trigonometric Functions
Trigonometric functions (sine, cosine, tangent) are used to relate angles and sides of right triangles. They provide a relationship between the angle and the ratio of two sides of the triangle:
- Sine (sin): sin(angle) = opposite side / hypotenuse
- Cosine (cos): cos(angle) = adjacent side / hypotenuse
- Tangent (tan): tan(angle) = opposite side / adjacent side
Identifying the Relevant Function
In this case, we are given the hypotenuse and the angle, and we need to find the opposite side (y). Therefore, the appropriate trigonometric function to use is sine (sin).
Applying Sine Function
We know:
To find 'y', we can rearrange the equation:
Calculating the Value of y
Using a calculator, we can find the sine of 48 degrees:
Now, we can substitute this value back into the equation:
- y ≈ 0.7431 * 10
- y ≈ 7.431
Rounding to the Nearest Hundredth
Rounding 7.431 to the nearest hundredth gives us:
Solution
Therefore, the value of y to the nearest hundredth is 7.43. This aligns with the provided answer choices, confirming our calculations.
Conclusion
By applying the sine function and using a calculator, we successfully determined the value of 'y' in the given right triangle. This demonstrates the importance of trigonometric functions in solving problems involving angles and sides of triangles. Understanding these concepts is crucial in various fields, including engineering, physics, and architecture.