In a triangle, if side a = 23, side b = 11 and angle A = 122 degrees, find the value of angle B to the nearest degree.
Find the value of angle A in the triangle where a = 8, b = 6 and c = 7 to the nearest degree.
Find the area, to the nearest whole number, of a triangle where a = 7 ft, b = 9.3 ft and angle C = 86 degrees.
Find the area of the triangle, to the nearest whole number, with a = 5, b = 6 and c = 9.
In a triangle, if side a = 23, side b = 11 and angle A = 122 degrees, find the value of angle B to the nearest degree.
find B using law of sines:
sinB/b = sinA/a
Find the value of angle A in the triangle where a = 8, b = 6 and c = 7 to the nearest degree.
find A using the law of cosines:
a^2 = b^2 + c^2 - 2bc cosA
Find the area, to the nearest whole number, of a triangle where a = 7 ft, b = 9.3 ft and angle C = 86 degrees.
area = 1/2 ab sinC
Find the area of the triangle, to the nearest whole number, with a = 5, b = 6 and c = 9.
use Heron's formula (much work), or find A using the law of cosines
and then the area is 1/2 bc sinA
To find the value of angle B in the first triangle, we can use the Law of Cosines. The formula is given by:
c^2 = a^2 + b^2 - 2ab*cos(C)
Substituting the given values into the formula:
23^2 = 11^2 + 122^2 - 2*11*122*cos(B)
529 = 121 + 14884 - 2684*cos(B)
Now, isolate cos(B) by rearranging the equation:
26864 - 529 - 14884 = 2684*cos(B)
11751 = 2684*cos(B)
Divide both sides by 2684:
cos(B) = 11751 / 2684
B = cos^(-1)(11751 / 2684)
To the nearest degree, angle B is approximately 144 degrees.
To find the value of angle A in the second triangle, we can use the Law of Cosines again. The formula is given by:
c^2 = a^2 + b^2 - 2ab*cos(C)
Substituting the given values into the formula:
7^2 = 8^2 + 6^2 - 2*8*6*cos(A)
49 = 64 + 36 - 96*cos(A)
Now, isolate cos(A) by rearranging the equation:
49 - 100 = -96*cos(A)
-51 = -96*cos(A)
Divide both sides by -96:
cos(A) = 51 / 96
A = cos^(-1)(51 / 96)
To the nearest degree, angle A is approximately 59 degrees.
To find the area of the third triangle, we can use the formula for the area of a triangle:
Area = (1/2) * b * c * sin(A)
Substituting the given values into the formula:
Area = (1/2) * 7 * 9.3 * sin(86)
Calculating the value:
Area = (1/2) * 7 * 9.3 * 0.996
Area = 32.772
To the nearest whole number, the area is 33 square feet.
To find the area of the fourth triangle, we can use Heron's formula:
s = (a + b + c) / 2
Area = sqrt(s * (s - a) * (s - b) * (s - c))
Substituting the given values into the formula:
s = (5 + 6 + 9) / 2 = 10
Area = sqrt(10 * (10 - 5) * (10 - 6) * (10 - 9))
Calculating the value:
Area = sqrt(10 * 5 * 4 * 1)
Area = sqrt(200)
Area ≈ 14.14
To the nearest whole number, the area is 14 square units.
To find the value of angle B in a triangle with given side lengths and angle measures, you can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
First, let's solve the first question.
Given:
a = 23
b = 11
angle A = 122 degrees
To find angle B, we can use the Law of Cosines since we know all three side lengths. Rearranging the formula, we get:
cos(B) = (a^2 + b^2 - c^2) / (2ab)
Substituting the given values:
cos(B) = (23^2 + 11^2 - c^2) / (2 * 23 * 11)
To calculate the value of B, we can now use the inverse cosine (arccos) function:
B = arccos((23^2 + 11^2 - c^2) / (2 * 23 * 11))
Please note that the given value of angle A is not required to find angle B.
Moving on to the next question:
Given:
a = 8
b = 6
c = 7
We need to find angle A. We can use the Law of Cosines again, but this time we will rearrange the formula to solve for angle A:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
Substituting the given values:
cos(A) = (6^2 + 7^2 - 8^2) / (2 * 6 * 7)
To find the value of angle A, we use the inverse cosine function:
A = arccos((6^2 + 7^2 - 8^2) / (2 * 6 * 7))
The third question involves finding the area of a triangle using the formula:
Area = (1/2) * a * b * sin(C)
Given:
a = 7 ft
b = 9.3 ft
angle C = 86 degrees
Substituting the given values:
Area = (1/2) * 7 * 9.3 * sin(86)
To calculate the area to the nearest whole number, we evaluate the expression using a calculator or mathematical software.
For the fourth question:
Given:
a = 5
b = 6
c = 9
To calculate the area of the triangle, we use Heron's formula:
s = (a + b + c) / 2
Area = sqrt(s * (s - a) * (s - b) * (s - c))
First, find the value of s:
s = (5 + 6 + 9) / 2 = 20 / 2 = 10
Substituting the values into the formula:
Area = sqrt(10 * (10 - 5) * (10 - 6) * (10 - 9))
To calculate the area to the nearest whole number, we evaluate the expression using a calculator or mathematical software.