In a triangle ABC, with angles A, B, and C and sides AB, BC, and AC, angle B is a right (90°) angle. If the sin of angle A is 0.5 and side BC is 8 inches long, what is the length of side AC?
Since angle B is a right angle, we can use the sine function to find the length of side AC.
Using the sine function, sin(A) = opposite/hypotenuse.
In this case, angle A is opposite side BC and hypotenuse is side AC.
sin(A) = BC/AC
0.5 = 8/AC
To find AC, we can cross multiply and solve for AC:
AC = 8/0.5
AC = 16 inches
Therefore, the length of side AC is 16 inches.
To find the length of side AC, we can use the Pythagorean theorem since angle B is a right angle.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
So, we have:
AC^2 = AB^2 + BC^2
Since angle B is a 90° angle, we know that AB and BC are the two sides adjacent to angle A.
Given that side BC is 8 inches long, we can substitute the values:
AC^2 = AB^2 + (8 inches)^2
Now, let's use the fact that the sine of angle A is 0.5 to find the length of side AB.
The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we know that the sine of angle A is 0.5, so:
sin(A) = opposite / hypotenuse
0.5 = AB / AC
Solving for AB, we have:
AB = 0.5 * AC
Now, we can substitute the value of AB in the equation for AC^2:
AC^2 = (0.5 * AC)^2 + (8 inches)^2
Expanding and simplifying:
AC^2 = 0.25 * AC^2 + 64 inches^2
Subtracting 0.25 * AC^2 from both sides:
AC^2 - 0.25 * AC^2 = 64 inches^2
Simplifying further:
0.75 * AC^2 = 64 inches^2
Dividing both sides by 0.75:
AC^2 = 64 inches^2 / 0.75
AC^2 = 85.333... inches^2
Taking the square root of both sides to find AC:
AC ≈ √(85.333...) inches
AC ≈ 9.244 inches (rounded to 3 decimal places)
Therefore, the length of side AC is approximately 9.244 inches.