Using the Law of Cosines, calculate the third side of a triangle if two sides are 10 and 14 units and they enclose an angle of 45 degrees.
user
Sure, I'd be happy to help you use the Law of Cosines to find the length of the third side.

The Law of Cosines states that for any triangle with sides a, b, and c, and an angle C opposite side c:

c^2 = a^2 + b^2 - 2ab cos C

Where cos C is the cosine of angle C, expressed in radians.

In your case, C is an acute angle (less than 90 degrees), so its measure in radians is given by:

C (in radians) = degrees * pi / 180

Substituting the given values, we have:

c^2 = 10^2 + 14^2 - 2 * 10 * 14 * cos(45 * pi/180)

Simplifying:

c^2 = 225 + 196 - 280 * cos(75pi/180)

* 75π/180 radians = 1.30903133 radians
  cos(1.30903133) = 0.173740993

So,

c^2 = 225 + 196 - 453.45 * 0.173740993

c^2 ≈ 296.59

Taking the square root, we get:

c ≈ √296.59 ≈ 16.66 units

Therefore, the length of the third side of the triangle, opposite the 45-degree angle and having sides of 10 and 14 units, is 16.66 units (rounded to two decimal places).
assistant
Using the Law of Cosines, calculate the third side of a triangle if two sides are 10 and 14 units and they enclose an angle of 45 degrees.
user
The Law of Cosines states that for a triangle with sides a, b, and c, and the angle C opposite side c, the following relationship holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our problem, we have:
a = 10 units
b = 14 units
C = 45 degrees

Now we can plug these values into the formula to find the length of side c:

c^2 = (10)^2 + (14)^2 - 2 * 10 * 14 * cos(45)
     = 100 + 196 - 280 * cos(45)

To proceed, we need to know the cosine of 45 degrees. The cosine of 45 degrees is approximately 0.7071 (you may use a calculator to find this value).

c^2 = 100 + 196 - 280 * 0.7071
     = 296 - 198.002
     ≈ 97.998

Now, to find the length of side c, we take the square root of both sides:

c ≈ √97.998
    ≈ 9.9 units

So, the length of the third side is approximately 9.9 units.
assistant