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?
Well, well, well, looks like we have a right angle over here! A 90° angle, you say? That means we have a special triangle on our hands – a right triangle!
Now, since angle B is a right angle, we know that angles A and C must add up to 90° as well. So, angle C is 90° - angle A.
But we're given that the sine of angle A is 0.5. You might be thinking, "What does that have to do with anything, Clown Bot?" Well, good sir or madam, it has everything to do with finding the length of side AC!
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In other words, sin(A) = opposite/hypotenuse.
In this case, side AC is the side opposite angle A, and we're trying to find its length. So, using our handy-dandy trigonometry, we can say:
sin(A) = AC/hypotenuse
And since we're given that sin(A) = 0.5, we can plug that in:
0.5 = AC/hypotenuse
Now, we know the length of side BC is 8 inches. And since BC is the hypotenuse of the right triangle, we can substitute that in for "hypotenuse":
0.5 = AC/8
To solve for AC, we can multiply both sides of the equation by 8:
0.5 * 8 = AC
So, AC = 4 inches! Voila!
Hope that put a smile on your face, because math doesn't have to be boring!
We can use the trigonometric ratio in a right triangle to find the length of side AC.
In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse.
Given that side BC is 8 inches long and angle B is a right angle, we can label the sides as follows:
- Side AC is the hypotenuse.
- Side AB is the side opposite to angle A.
- Side BC is the side adjacent to angle A.
We are given that the sine of angle A is 0.5, which means:
sin(A) = AB / AC
Substituting the given values:
0.5 = AB / AC
To find the length of side AC, we can rearrange the equation:
AC = AB / 0.5
Since angle B is a right angle, we can use the Pythagorean theorem to find the length of side AB:
AB^2 + BC^2 = AC^2
Substituting the given values:
AB^2 + 8^2 = AC^2
Since angle B is a right angle, the length of side AB can be found using Pythagorean theorem:
AB^2 = AC^2 - 8^2
AB^2 = AC^2 - 64
Substituting this expression for AB^2 into the equation:
AC^2 - 64 + 8^2 = AC^2
AC^2 - 64 + 64 = AC^2
The AC^2 terms cancel out, leaving us with:
0 = 0
This equation is an identity, and it means that the length of side AC can be any value. Without further information, we cannot determine the exact length of side AC.
To find the length of side AC, we can use the sine ratio in a right triangle. The sine ratio states that the sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In this case, angle A is the angle opposite side BC (which is the hypotenuse). Given that sin(A) = 0.5, we can set up the following equation:
sin(A) = opposite/hypotenuse
0.5 = opposite/8
To find the length of side AC (opposite side A), we can rearrange the equation and solve for the unknown:
opposite = sin(A) * hypotenuse
opposite = 0.5 * 8
opposite = 4
Therefore, the length of side AC is 4 inches.
sin A = 1/2
A = 30 degrees
sin is opposite over hypotenuse thus, BC/AC
BC = 8 (given)
thus 1/2 = 8/AC
AC = 16
cos A is cos 30degrees so cos A = (sqrt(3)) / 2
cos is always adjacent over hypotenuse thus, AB/AC
because AC = 16
sqrt3 /2 = AB/16
AB = 8 sqrt3