The smaller two sides of an obtuse triangle are known to be 4.0 m and 10.6 m. The angle that is opposite 10.6 is 14 degrees.
A) Calculate the obtuse angle in the triangle, to the nearest tenth of a degree. Show all your work and make sure you include the equation.
In ∆ABC, let A=14°, a=10.6, c=4
We want to find B.
sinC/c = sinA/a, so sinC/4 = sin14°/10.6
sinC = 0.09129
C = 5.23°
So B = 180-(14-5.23) = 160.76°
To solve for the obtuse angle in the triangle, we can use the Law of Cosines. The Law of Cosines is given by the equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where c represents the side opposite angle C, and a and b represent the other two sides of the triangle.
In this case, we know that side a is 4.0 m, side b is 10.6 m, and angle C is 14 degrees.
First, let's plug in the known values into the equation:
c^2 = 4.0^2 + 10.6^2 - 2 * 4.0 * 10.6 * cos(14)
Simplifying this equation further:
c^2 = 16 + 112.36 - 84.8 * cos(14)
Next, let's evaluate the cosine of 14 degrees. You can use a scientific calculator to find the value of cos(14). In this case, cos(14) is approximately 0.9703.
Now, we substitute this value into the equation:
c^2 = 16 + 112.36 - 84.8 * 0.9703
Simplifying further:
c^2 = 16 + 112.36 - 82.23544
c^2 ≈ 46.12456
To find c, we take the square root of both sides:
c ≈ √46.12456
Finally, we can calculate the obtuse angle using the Law of Sines. The Law of Sines is given by the equation:
sin(C)/c = sin(A)/a
In this case, we know side c is approximately √46.12456, which we just calculated, and side a is 4.0 m. We also know that angle C is 14 degrees. We can solve for angle A using this equation.
sin(14)/√46.12456 = sin(A)/4.0
To find angle A, multiply both sides by 4.0:
sin(14) = 4.0 * sin(A) / √46.12456
Now, we solve for sin(A). Divide both sides by 4.0 and multiply by √46.12456:
sin(A) = sin(14) * √46.12456 / 4.0
Using a scientific calculator, calculate the sine of 14 degrees, which is approximately 0.2419.
Now, substitute this value into the equation:
sin(A) ≈ 0.2419 * √46.12456 / 4.0
Solve the right side of the equation using a calculator:
sin(A) ≈ 0.2419 * 6.7892 / 4.0
sin(A) ≈ 0.4099
Now, to find the obtuse angle A, we take the inverse sine (also known as arcsine) of 0.4099 using a calculator:
A ≈ arcsin(0.4099)
A ≈ 24.8 degrees
The obtuse angle in the triangle is approximately 24.8 degrees.
To calculate the obtuse angle in the triangle, we can use the Law of Cosines. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c is the side opposite angle C
a and b are the other two sides
C is the angle opposite side c
We are given:
a = 4.0 m
b = 10.6 m
C = 14 degrees
Let's plug in the values we have into the equation:
c^2 = 4.0^2 + 10.6^2 - 2(4.0)(10.6) * cos(14)
Simplifying:
c^2 = 16.0 + 112.36 - 84.8 * cos(14)
c^2 = 128.36 - 84.8 * cos(14)
To find c, we take the square root of both sides:
c = √(128.36 - 84.8 * cos(14))
Calculating the value:
c ≈ √(128.36 - 84.8 * cos(14)) ≈ √(128.36 - 84.8 * 0.9703) ≈ √(128.36 - 82.27444) ≈ √46.08556 ≈ 6.78
We have now calculated the value of the side c. To find the obtuse angle, we can use the Law of Cosines again:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Plugging in the values we have:
cos(C) = (4.0^2 + 10.6^2 - 6.78^2) / (2 * 4.0 * 10.6)
Simplifying:
cos(C) = (16.0 + 112.36 - 46.1524) / 84.8
cos(C) = 82.2076 / 84.8
cos(C) ≈ 0.9692
To find the obtuse angle, we take the inverse cosine of the value we just calculated:
C ≈ cos^(-1)(0.9692)
Using a calculator, we find:
C ≈ 14.2 degrees
Therefore, the obtuse angle in the triangle is approximately 14.2 degrees to the nearest tenth of a degree.