A local naturalist says that a triangular area on the north side of a pond is a turtle habitat. The lengths of the sides of this area are 15 meters, 25 meters, and 36 meters. Which expression gives the measure of the angle between the sides of the habitat that measure 15 meters and 25 meters?
If the angle is θ, the law of cosines says that
36^2 = 15^2 + 25^2 - (2)(15)(25) cosθ
To find the measure of the angle between the sides that measure 15 meters and 25 meters, we need to use the concept of trigonometry. Specifically, we can use the Law of Cosines to solve this problem.
The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of the two sides multiplied by the cosine of the included angle. This can be represented by the equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c represents the length of the side opposite to angle C, and a and b represent the lengths of the other two sides.
In this case, we have a triangle with sides of lengths 15 meters, 25 meters, and 36 meters. Let's assign these values to the variables a, b, and c:
a = 15 meters
b = 25 meters
c = 36 meters
Our goal is to find the measure of angle C, which is the angle between the sides that measure 15 meters and 25 meters.
Now, we can use the Law of Cosines equation to solve for the cosine of angle C:
c^2 = a^2 + b^2 - 2ab * cos(C)
Plugging in the values we have:
36^2 = 15^2 + 25^2 - 2 * 15 * 25 * cos(C)
Simplifying:
1296 = 225 + 625 - 750 * cos(C)
Now, we can rearrange the equation to isolate cos(C):
-554 = -750 * cos(C)
Dividing both sides of the equation by -750:
cos(C) = -554 / -750
cos(C) ≈ 0.739
To find the measure of angle C, we can use the inverse cosine function (cos^-1 or arccos):
C ≈ cos^-1(0.739)
Using a calculator or a trigonometric table, we find that the approximate value of C is 42.6 degrees.
Therefore, the angle between the sides that measure 15 meters and 25 meters in the turtle habitat is approximately 42.6 degrees.
So, the expression that gives the measure of this angle would be: 42.6 degrees.