Question

Consider a triangle where a = 3, b = 5, and c = 7. Use the law of cosines to find the angle measure c in degrees. a) 36.87 degrees b) 53.13 degrees c) 90 degrees d) 120 degrees

Answer 1

To find the angle measure C in the given triangle, we can use the law of cosines.

The law of cosines states that in a triangle with sides of lengths a, b, and c, and opposite angles A, B, and C respectively, the following equation holds:

c^2 = a^2 + b^2 - 2abcos(C)

Given that a = 3, b = 5, and c = 7, we can substitute these values into the equation:

7^2 = 3^2 + 5^2 - 2(3)(5)cos(C)

49 = 9 + 25 - 30cos(C)

49 = 34 - 30cos(C)

Subtracting 34 from both sides of the equation gives:

15 = -30cos(C)

Dividing both sides by -30 gives:

-0.5 = cos(C)

To find the angle measure C, we need to take the inverse cosine (or arccosine) of -0.5:

C = arccos(-0.5)

Using a calculator, we find that the angle measure C is approximately 120 degrees.

Therefore, the correct answer is d) 120 degrees.

Answer 2

The solution to this problem involves using the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For triangle with sides a, b, c and angle C between sides a and b, the relation is given by:

cos(C) = (a² + b² - c²) / 2ab

We plug our values into the formula, with a = 3, b = 5, and c = 7:

cos(C) = (3² + 5² - 7²) / (2 * 3 * 5)

which simplifies to

cos(C) = (-0.5)

So the cosine of angle C is -0.5. To find the angle, we use the arccos function (also written as acos, or "cos^-1"), which gives us the angle that corresponds to a given cosine.

So, acos(-0.5) gives us a result about 2.094 radians.

However, the problem asks for the angle in degrees, not radians. To convert from radians to degrees, we multiply the angle in radians by the conversion factor of 180/π.

Our angle, therefore, is 2.094 * (180 / π) which is approximately 120 degrees.

So, the correct answer is (d) 120 degrees.

Answer: d) 120 degrees