the longest side of a triangle is 34 feet. the measures of two angles of the triangle are 40 and 65 degrees. find the lengths of the other two sides using law of sines and cosines
the longest side of a triangle is across from the largest angle.
http://library.thinkquest.org/C0121962/sincoslaws.htm
To find the lengths of the other two sides of a triangle using the Law of Sines and Cosines, we need to know either the length of one side and the measures of two angles or the lengths of two sides and the measure of the included angle. In this case, we know the length of the longest side (34 feet) and the measures of two angles (40 and 65 degrees).
Let's begin by using the Law of Sines to find the lengths of the other two sides. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle.
1. Start by finding the measure of the third angle. Since the sum of the angles in a triangle is 180 degrees, we can subtract the given angles from 180:
Third angle = 180 - 40 - 65 = 75 degrees.
2. Now, using the Law of Sines, we can write the following equation:
sin(40)/34 = sin(75)/x
Here, x represents the length of the side opposite the angle of 40 degrees, which we want to find.
3. Rearrange the equation to solve for x. Cross-multiply and isolate x:
x = (34 * sin(40)) / sin(75)
Using a scientific calculator, evaluate sin(40) and sin(75):
x ≈ (34 * 0.6428) / 0.9659
x ≈ 22.03
Therefore, the length of the side opposite the angle of 40 degrees is approximately 22.03 feet.
4. To find the length of the remaining side, we can use the fact that the sum of the measures of the angles in a triangle is 180 degrees. Therefore, the sum of the other two angles is 180 - 65 - 75 = 40 degrees.
Additionally, we know that the sine of an angle is equal to the sine of its supplement (angle that adds up to 180 degrees), so sin(40) = sin(180 - 40).
5. Using the Law of Sines again, we can write the following equation:
sin(40) / y = sin(180 - 40) / 34
Here, y represents the length of the side opposite the angle of 65 degrees.
6. Rearrange the equation to solve for y. Cross-multiply and isolate y:
y = (34 * sin(180 - 40)) / sin(40)
Using a scientific calculator, evaluate sin(180 - 40) and sin(40):
y ≈ (34 * 0.866) / 0.6428
y ≈ 45.97
Therefore, the length of the side opposite the angle of 65 degrees is approximately 45.97 feet.
In conclusion, using the Law of Sines, we found that the length of the side opposite the angle of 40 degrees is approximately 22.03 feet, and the length of the side opposite the angle of 65 degrees is approximately 45.97 feet.