Find the area of the triangle ABC given A = 30°, b = 5, and c = 2.
Use the law of cosines to find side a.
a^2=b^2+c^2 -2bcCosA
Then area= sqrt [s(s-a)(s-b)(s-c)]
where s is half the perimeter.
There are about 100 other ways to work this.
To find the area of triangle ABC, we can use the formula for the area of a triangle given two sides and the included angle. Let's call the included angle A, the side opposite angle A as a, the side opposite angle B as b, and the side opposite angle C as c. The formula for the area of a triangle is:
Area = (1/2) * a * b * sin(A)
Given A = 30°, b = 5, and c = 2, we can find side a using the law of sines. The law of sines states that for any triangle ABC:
sin(A) / a = sin(B) / b = sin(C) / c
Since A = 30° and b = 5, we can rearrange the equation to solve for a:
sin(30°) / a = sin(B) / 5
By substituting the values, we have:
sin(30°) / a = sin(B) / 5
(1/2) / a = sin(B) / 5
a = (5 * sin(B)) / 2
Now, let's find angle B using the fact that the sum of the angles in a triangle is 180°. Since we know angle A = 30°, we can find angle B:
B = 180° - A - C
B = 180° - 30° - 90° (since C = 90° in a right-angled triangle)
B = 60°
Using the values we found, we can calculate side a:
a = (5 * sin(B)) / 2
a = (5 * sin(60°)) / 2
a = (5 * √3/2) / 2
Now we can substitute the values of a, b, and A into the formula for the area of a triangle:
Area = (1/2) * a * b * sin(A)
Area = (1/2) * [(5 * √3/2) / 2] * 5 * sin(30°)
Calculating the values:
Area = (1/2) * [(5 * √3/2) / 2] * 5 * 1/2
Area = (1/2) * (5 * 3/2) * 5 * 1/2
Area = (1/2) * (15/2) * 5 * 1/2
Area = (1/2) * (15/2) * 5/2
Area = (1/2) * 15/2 * 5/2
Area = (1/2) * 15/2 * 5/2
Area = (1/2) * 15 * 5 / 4
Area = (15 * 5) / (2 * 4)
Area = 75 / 8
Therefore, the area of triangle ABC is 75/8 or 9.375 units squared.
To find the area of a triangle, we can use the formula:
Area = (1/2) * base * height.
In this case, the given values are A = 30°, b = 5, and c = 2. However, we need the height of the triangle in order to calculate the area.
Given the angle A, we can use the sine function to find the height of the triangle. The formula for the height h is:
h = b * sin(A).
Plugging in the values, we have:
h = 5 * sin(30°).
First, we need to convert the angle from degrees to radians because the trigonometric functions in most programming languages expect the input in radians. We know that 180° = π radians, so we can convert as follows:
30° = (30/180) * π = π/6 radians.
Now we can calculate the height:
h = 5 * sin(π/6).
Using a scientific calculator or trigonometric functions in a programming language, we find that sin(π/6) is equal to 0.5. Therefore, the height is:
h = 5 * 0.5 = 2.5.
Now that we have the height, we can calculate the area of the triangle:
Area = (1/2) * base * height = (1/2) * 2 * 2.5 = 2.5.
Therefore, the area of triangle ABC is 2.5 square units.