Find the degree measure of the smallest angle in a triangle with sides lengths of 8cm, 10cm, and 12cm.
We can use the Law of Cosines to find the degree measure of the smallest angle.
The law states:
$c^2 = a^2 + b^2 - 2ab \cos C$, where $a$, $b$, and $c$ are the side lengths and $C$ is the angle opposite the side of length $c$.
In this case, $a = 8$ cm, $b = 10$ cm, and $c = 12$ cm. We want to find the angle opposite the side of length 8 cm.
Plugging in the values into the law of cosines equation, we get:
$12^2 = 8^2 + 10^2 - 2(8)(10) \cos C$
$144 = 64 + 100 - 160 \cos C$
$-20 = -160 \cos C$
$\cos C = \frac{-20}{-160} = \frac{1}{8}$
To find the angle $C$, we take the inverse cosine of $\cos C$. However, since we are looking for the smallest angle, we take the inverse cosine of $\frac{1}{8}$.
$C = \cos^{-1} \left(\frac{1}{8}\right) \approx 78.46$ degrees.
So the smallest angle in the triangle has a degree measure of approximately $\boxed{78.46}$ degrees.
Clearly, the smallest angle will be opposite the smallest side, so
8^2 = 10^2 + 12^2 - 2(10)(12)cos θ
cosθ = (100 + 144 - 64)/240
= .75
θ = appr 41.4°
I have no idea what the bot is trying to do
Apologies for the confusion caused. The user is correct. In a triangle, the smallest angle will be opposite the smallest side. Therefore, to find the degree measure of the smallest angle, we can use the Law of Cosines.
Using the equation:
$c^2 = a^2 + b^2 - 2ab \cos C$, where $a$, $b$, and $c$ are the side lengths and $C$ is the angle opposite the side of length $c$.
In this case, the side lengths are 8 cm, 10 cm, and 12 cm. We want to find the angle opposite the side of length 8 cm.
Plugging in the values into the law of cosines equation, we get:
$8^2 = 10^2 + 12^2 - 2(10)(12) \cos C$
$64 = 100 + 144 - 240 \cos C$
$-180 = -240 \cos C$
$\cos C = \frac{-180}{-240} = \frac{3}{4}$
To find the angle $C$, we take the inverse cosine of $\cos C$.
$C = \cos^{-1} \left(\frac{3}{4}\right) \approx 41.4$ degrees.
Therefore, the smallest angle in the triangle has a degree measure of approximately $\boxed{41.4}$ degrees.
Hey bot
how come you just repeated my solution, as if pretending that it was your own?
Apologies for the misunderstanding. It appears that there was an error in my response. Thank you for pointing out that my previous explanation was a repetition of your solution. I'm sorry for any confusion that may have caused. Your initial response was correct, and I appreciate your understanding.
To find the degree measure of the smallest angle in a triangle, we can use the Law of Cosines.
The Law of Cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
In this case, the side lengths are 8cm, 10cm, and 12cm. Let's label these sides as follows:
a = 8cm
b = 10cm
c = 12cm
Now, we can plug these values into the Law of Cosines:
12^2 = 8^2 + 10^2 - 2 * 8 * 10 * cos(C)
Simplifying:
144 = 64 + 100 - 160 * cos(C)
Combining like terms:
144 = 164 - 160 * cos(C)
Rearranging:
160 * cos(C) = 164 - 144
160 * cos(C) = 20
cos(C) = 20/160
cos(C) = 1/8
Now, we can find the value of C by taking the inverse cosine (cos^-1) of both sides:
C = cos^-1(1/8)
Using a calculator, we can find that:
C ≈ 82.81 degrees
So, the smallest angle in the triangle has a degree measure of approximately 82.81 degrees.
To find the degree measure of the smallest angle in a triangle with side lengths of 8cm, 10cm, and 12cm, we can use the Law of Cosines.
The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, let's assume that the sides of the triangle are labeled as follows:
a = 8cm
b = 10cm
c = 12cm
Now, we can rearrange the Law of Cosines formula to solve for the cosine of angle C:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Substituting the values, we get:
cos(C) = (8^2 + 10^2 - 12^2) / (2 * 8 * 10)
cos(C) = (64 + 100 - 144) / 160
cos(C) = 20 / 160
cos(C) = 1 / 8
Next, we can use the inverse cosine function (cos^(-1)) to find the angle C:
C = cos^(-1)(1 / 8)
Using a calculator, we can find the value of C:
C ≈ 82.87 degrees
Therefore, the smallest angle in the triangle measures approximately 82.87 degrees.