Given ABC with b = 21, c = 32, and mA 40° , find a. Round the cosine value to the nearest thousandth and answer to the nearest hundredth.
In this one I got 22.43
a^2 = b^2 + c^2 - 2bc cosA
a^2 = 21^2 + 32^2 - 2*21*32*cos40°
a^2 = 435.4
a = 20.86
So, how did you get 22.43 ?
Oh... I just checked and I did it wrong
To find side length a of triangle ABC, we can use the Law of Cosines, which states that in any triangle ABC with side lengths a, b, and c, and angle A opposite side a, the following equation holds:
a^2 = b^2 + c^2 - 2bc*cosA
Plugging in the given values:
b = 21
c = 32
mA = 40°
Let's calculate the cosine of angle A:
cosA = cos(40°)
To find the value of cos(40°), we can use a calculator or a trigonometric table.
cos(40°) ≈ 0.766
Now we can substitute the values into the Law of Cosines equation:
a^2 = 21^2 + 32^2 - 2*21*32*0.766
Simplifying the equation:
a^2 ≈ 441 + 1024 - 1300.608
a^2 ≈ 165.392
To find a, we need to solve for a:
a ≈ √165.392
Using a calculator or rounding to the nearest hundredth:
a ≈ 12.86
Therefore, the length of side a is approximately 12.86.
To find side a in triangle ABC, we can use the Law of Cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides times the cosine of the enclosed angle.
The formula for the Law of Cosines is:
a^2 = b^2 + c^2 - 2bc * cos(A)
Given that b = 21, c = 32, and mA = 40°, we can now substitute the values into the formula:
a^2 = 21^2 + 32^2 - 2 * 21 * 32 * cos(40°)
To calculate the value of cos(40°), you can use a scientific calculator or the math function on your phone or computer. Rounding the cosine value to the nearest thousandth would be sufficient.
Once you have calculated the value of cos(40°), substitute it back into the formula:
a^2 = 21^2 + 32^2 - 2 * 21 * 32 * (cosine value)
Finally, take the square root of both sides to solve for a:
a = sqrt(a^2)
Calculate the value and round it to the nearest hundredth, as asked in the question.
By following these steps, you should be able to find the value of side a in triangle ABC.