in dxy2,y=29.8,2=51.4,x=19.6cm.solve the triangle completely
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I will guess that "dxy2" means ∆XYZ, but you sure don't make things easy.
z^2 = x^2 + y^2 - 2xy cosZ = 19.6^2 + 29.8^2 - 2*19.6*29.8 cos51.4°
z = 23.3
sinX/x = sinZ/z
sinX = 19.6/23.3 sin51.4°
X = 41.1°
Z = 180-51.4-41.1 = 87.5°
To solve the triangle completely in the given scenario, we can use the Law of Cosines and Law of Sines.
1. Start by finding the missing side, d, using the Law of Cosines:
c² = a² + b² - 2ab * cos(C)
Given: a = 29.8 cm, b = 51.4 cm, C = 90° (as it is a right triangle)
We need to find c.
c² = (29.8)² + (51.4)² - 2(29.8)(51.4) * cos(90°)
c² = 888.04 + 2641.96 - 0
c² = 3530
c = √(3530)
c ≈ 59.4 cm
Therefore, the missing side, d, is approximately 59.4 cm.
2. Next, use the Law of Sines to find the remaining angles, A and B:
sin(A) / a = sin(B) / b = sin(C) / c
Given: a = 29.8 cm, b = 51.4 cm, c ≈ 59.4 cm
sin(A) / 29.8 = sin(90°) / 59.4
sin(A) = (29.8 * sin(90°)) / 59.4
sin(A) ≈ 0.5
A ≈ arcsin(0.5)
A ≈ 30°
sin(B) / 51.4 = sin(90°) / 59.4
sin(B) = (51.4 * sin(90°)) / 59.4
sin(B) ≈ 0.87
B ≈ arcsin(0.87)
B ≈ 60°
Therefore, angle A is approximately 30° and angle B is approximately 60°.
3. Finally, calculate the remaining angle, angle C:
C = 180° - A - B
C = 180° - 30° - 60°
C = 90°
Therefore, angle C is exactly 90°.
In summary, based on the given values, the triangle is completely solved, with side a = 29.8 cm, side b = 51.4 cm, side c ≈ 59.4 cm, and angles A ≈ 30°, B ≈ 60°, and C = 90°.
To solve the triangle completely, we need to find all the missing angles and sides of the triangle.
Given:
dxy2, y = 29.8 cm (side opposite angle Y)
dxy2, 2 = 51.4 cm (side opposite angle 2)
dxy2, x = 19.6 cm (side opposite angle X)
Step 1: Use the Law of Cosines to find the missing angle.
The Law of Cosines states that for any triangle with sides a, b, c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
We can use this formula to find the missing angle. Let's solve for angle X first.
c = dxy2, x = 19.6 cm
a = dxy2, y = 29.8 cm
b = dxy2, 2 = 51.4 cm
19.6^2 = 29.8^2 + 51.4^2 - 2 * 29.8 * 51.4 * cos(X)
382.16 = 884.04 + 2643.96 - 3062.92 * cos(X)
-2534.84 = -3062.92 * cos(X)
cos(X) = -2534.84 / -3062.92
cos(X) = 0.8268
Now, we can use the inverse cosine function (cos^(-1) or arccos) to find angle X:
X = cos^(-1)(0.8268)
Using a calculator, X ≈ 33.58 degrees
Step 2: Use the Law of Sines to find another angle.
The Law of Sines states that for any triangle with sides a, b, c, and angles A, B, and C opposite their respective sides, the following equation holds:
a/sin(A) = b/sin(B) = c/sin(C)
We can use this formula to find angle Y.
a = dxy2, y = 29.8 cm
b = dxy2, x = 19.6 cm
A = angle X = 33.58 degrees
29.8/sin(Y) = 19.6/sin(33.58)
sin(Y) = (29.8 * sin(33.58)) / 19.6
sin(Y) ≈ 0.7148
Now, we can use the inverse sine function (sin^(-1) or arcsin) to find angle Y:
Y = sin^(-1)(0.7148)
Using a calculator, Y ≈ 44.17 degrees
Step 3: Find angle Z.
To find angle Z, we can use the fact that the sum of the angles in a triangle is always 180 degrees:
Z = 180 - X - Y
Z = 180 - 33.58 - 44.17
Z ≈ 102.25 degrees
Now we have found all the angles of the triangle.
Step 4: Find the remaining side lengths using the Law of Sines.
We already have the sides dxy2, y = 29.8 cm, dxy2, x = 19.6 cm, and dxy2, 2 = 51.4 cm.
To find the remaining side length, we can use the Law of Sines:
dxy2, 1/sin(X) = 29.8/sin(Z)
dxy2, 1 = (29.8 * sin(X)) / sin(Z)
Using the values we have calculated:
dxy2, 1 = (29.8 * sin(33.58)) / sin(102.25)
Using a calculator, dxy2, 1 ≈ 9.72 cm
Now we have found all the side lengths and angles of the triangle. The triangle is completely solved.