To the nearest minute, what is the approximate size of the smallest angle of a triangle whose sides are 4,5, and 8?
How do I find the angle?
Use the cosine law.
4^2 = 5^2 + 8^2 - 2(5)(8)cos ß
where ß is the angle opposite the side of 4
To find the smallest angle of a triangle, you can use the Law of Cosines. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c represents the length of the side opposite the angle you want to find (in this case, the smallest angle).
a and b represent the lengths of the other two sides of the triangle.
C represents the angle opposite side c.
In this case, you have a triangle with sides of 4, 5, and 8. Let's assume that side c is opposite the smallest angle. Then, we can plug in the values into the Law of Cosines formula and solve for cos(C):
8^2 = 4^2 + 5^2 - 2 * 4 * 5 * cos(C)
64 = 16 + 25 - 40 * cos(C)
64 = 41 - 40 * cos(C)
40 * cos(C) = 41 - 64
40 * cos(C) = -23
Now we can solve for cos(C):
cos(C) = -23/40
To find the smallest angle, you will need to find the inverse cosine (also known as arccos or cos^-1) of -23/40. On most calculators, this function is denoted as "cos^-1" or "arccos." Using a calculator, find the inverse cosine of -23/40:
cos^-1(-23/40) ≈ 122.49°
To convert degrees to minutes, multiply the decimal part by 60:
0.49 * 60 ≈ 29.4
Therefore, the approximate size of the smallest angle in the triangle is 122 degrees and 29 minutes.
To find the smallest angle of a triangle given the lengths of its sides, we can use the Law of Cosines. According to the Law of Cosines, if we have a triangle with sides of lengths a, b, and c, the cosine of one of the angles, let's call it angle A, can be found using the following formula:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
In this case, the sides of the triangle are 4, 5, and 8. Let's label them as follows:
a = 4
b = 5
c = 8
We want to find the smallest angle, so we need to find the angle whose side is opposite to the side of length 8. Let's call it angle A.
Using the Law of Cosines formula, we have:
cos(A) = (5^2 + 8^2 - 4^2) / (2 * 5 * 8)
cos(A) = (25 + 64 - 16) / 80
cos(A) = 73 / 80
Now, to find angle A, we can use the inverse cosine function (cos^(-1)) or arccosine. This function gives us the angle whose cosine is equal to the given value.
A = cos^(-1)(73/80)
Using a calculator or a tool with a trigonometric function, we can find that A ≈ 19.178 degrees.
Since we want to find the size of the angle to the nearest minute, we can multiply the decimal value of the angle by 60 to convert it to minutes.
19.178 degrees * 60 minutes/degree ≈ 1151 minutes
Therefore, the approximate size of the smallest angle in the triangle whose sides are 4, 5, and 8 is approximately 1151 minutes.