Answer:
the possible sizes of angle A in triangle ABC are 60° and 120°.
Explanation:
1. To prove that in any triangle ABC, the area of triangle ABC is equal to 1/2 times the product of sides b and c, multiplied by the sine of angle A, we can use the formula for the area of a triangle.
The formula for the area of a triangle is given as:
Area = (1/2) * base * height
In triangle ABC, we can take side b as the base and drop a perpendicular from vertex A to side b, forming the height of the triangle.
Let's call the height of the triangle h. Now, we can express the area of triangle ABC as:
Area = (1/2) * b * h
To find the height of the triangle, we can use the sine of angle A. The sine of an angle is defined as the ratio of the opposite side to the hypotenuse in a right triangle.
In triangle ABC, we have:
sin A = h / c
Rearranging the equation, we get:
h = c * sin A
Now, substituting the value of h in the formula for the area, we have:
Area = (1/2) * b * (c * sin A)
Simplifying the expression, we get:
Area = (1/2) * b * c * sin A
Therefore, we have proven that in any triangle ABC, the area of triangle ABC is equal to 1/2 times the product of sides b and c, multiplied by the sine of angle A.
2. To determine the area of triangle PQR, we can use the formula derived in the previous question:
Area = (1/2) * b * c * sin A
Given p = 3.7, q = 5.2, and R = 112°, we can substitute these values into the formula.
Let's assign p to side b, q to side c, and R to angle A.
b = p = 3.7
c = q = 5.2
A = R = 112°
Now, we can calculate the area using the formula:
Area = (1/2) * p * q * sin R
Substituting the given values, we have:
Area = (1/2) * 3.7 * 5.2 * sin 112°
Calculating sin 112° using trigonometric tables or a calculator, we get:
Area ≈ 8.023 square units (rounded to three decimal places)
Therefore, the area of triangle PQR is approximately 8.023 square units.
3. To determine the possible sizes of angle A, given the area of triangle ABC as 6√3 m², b = 8 m, and c = 3 m, we can use the formula derived in the first question:
Area = (1/2) * b * c * sin A
Given Area = 6√3 m², b = 8 m, and c = 3 m, we can substitute these values into the formula.
Let's assign b = 8, c = 3, and Area = 6√3.
b = 8
c = 3
Area = 6√3
Now, we can calculate the sine of angle A using the formula:
Area = (1/2) * b * c * sin A
Substituting the given values, we have:
6√3 = (1/2) * 8 * 3 * sin A
Simplifying the equation, we get:
6√3 = 12 * sin A
Dividing both sides by 12, we have:
√3/2 = sin A
To determine the possible values of angle A, we need to find the inverse sine (arcsin) of √3/2.
Using trigonometric tables or a calculator, we find that the possible values of A are:
A = 60° or A = 120°
Therefore, without using a calculator, the possible sizes of angle A in triangle ABC are 60° and 120°.