What is sinB ?
Express your answer as a fraction.
The Correct Answer and Explanation is :
To find (\sin B) and express it as a fraction, we typically use a right-angled triangle and the definition of sine, which is the ratio of the length of the side opposite angle (B) to the length of the hypotenuse.
Here’s how you can solve for (\sin B) using an example problem:
Problem:
Given a right triangle where the hypotenuse is (17) units and one leg is (15) units, find (\sin B).
Step-by-step Solution:
- Identify the sides of the triangle:
- Hypotenuse ((c)) = (17)
- One leg ((a)) = (15)
- The other leg ((b)) is unknown.
- Use the Pythagorean theorem to find the missing side:
[
a^2 + b^2 = c^2
]
Substituting the known values:
[
15^2 + b^2 = 17^2
]
[
225 + b^2 = 289
]
Solving for (b^2):
[
b^2 = 289 – 225 = 64
]
Taking the square root of both sides:
[
b = 8
]
So, the length of the side opposite angle (B) is (8). - Calculate (\sin B):
The sine of an angle in a right triangle is given by:
[
\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{17}
]
Conclusion:
The value of (\sin B) is (\frac{8}{17}). This fraction represents the ratio of the side opposite angle (B) to the hypotenuse in this triangle.
This method of finding (\sin B) works for any right triangle as long as you know the lengths of two sides, and it relies on the basic properties of triangles and trigonometric ratios.