A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What is the measure to the nearest degree of the smallest angle in the frame? (Enter only the number.)
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To find the measure of the smallest angle in the triangular frame, we can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the length of the side opposite angle C
- a and b are the lengths of the other two sides
In this case, let's assume that the side with length 20 cm is opposite to angle A, the side with length 24 cm is opposite to angle B, and the side with length 30 cm is opposite to angle C (the smallest angle).
Using the formula, we have:
30^2 = 20^2 + 24^2 - 2 * 20 * 24 * cos(C)
900 = 400 + 576 - 960 * cos(C)
900 = 976 - 960 * cos(C)
960 * cos(C) = 76
cos(C) = 76/960
Now, we can use the inverse cosine (cos^-1) function to find the measure of angle C.
C = cos^-1(76/960) ≈ 81.9 degrees (rounded to one decimal place)
Therefore, the measure of the smallest angle in the frame is approximately 81.9 degrees.
To find the smallest angle in the triangular frame, we can use the Law of Cosines. According to the Law of Cosines, the squared length of one side of a triangle is equal to the sum of the squares of the other two sides minus two times the product of the lengths of those two sides multiplied by the cosine of the included angle.
Let's label the sides of the triangle as follows:
Side a = 20 cm
Side b = 24 cm
Side c = 30 cm
To find the smallest angle, we need to determine the side length opposite to it. Let's call it side x.
Using the Law of Cosines, we have:
x^2 = a^2 + b^2 - 2ab * cos(C)
Where C is the included angle (opposite to side c).
Let's calculate:
x^2 = 20^2 + 24^2 - 2(20)(24) * cos(C)
x^2 = 400 + 576 - 960 * cos(C)
x^2 = 976 - 960 * cos(C)
Now, we can solve for x:
x ≈ √(976 - 960 * cos(C))
To find the measure of the smallest angle, we need to find the value of cos(C). We can use the Law of Cosines once again:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
cos(C) = (20^2 + 24^2 - 30^2) / (2 * 20 * 24)
cos(C) = (400 + 576 - 900) / 960
cos(C) = 76 / 960
cos(C) ≈ 0.0791
Now, we can substitute this value back into the equation for x:
x ≈ √(976 - 960 * 0.0791)
x ≈ √(976 - 75.84)
x ≈ √900.16
x ≈ 30.001
Therefore, the length of the opposite side (x) is approximately 30.001 cm.
Now, we can find the smallest angle by using the inverse cosine function:
Smallest angle ≈ cos^(-1)(0.0791)
Smallest angle ≈ 85.267 degrees
Rounded to the nearest degree, the smallest angle in the frame is approximately 85 degrees.
law of sines: smallest angle opposite smallest side.
That would be between the sides of 24 and 30. The law of cosines tells us that
20^2 = 24^2 + 30^2 - 2(24)(30) cosθ
Now just do the math