Calculate the value of a, A and C in ∆Abc given that b = 17.23cm, c= 10.86cm and B= 101°15
use the law of sines.
sinC/c = sinB/b
sinC/10.86 = sin101°15'/17.23
C = 38.18°
unless you meant C=101.15°, in which case you need to adjust this result.
Then it is easy to find A, since A+B+C=180°
Then, using the law of sines again, a/sinA = b/sinB
or, using the law of cosines, a^2 = b^2+c^2 - 2bc cosA
Using the law of cosine
To solve for the value of angles and side lengths in a triangle, we can use the Law of Sines and Law of Cosines.
First, let's calculate the value of angle A using the Law of Sines:
sin(A) / a = sin(B) / b
We are given:
B = 101°15°
b = 17.23cm
Plugging in these values, we get:
sin(A) / a = sin(101°15') / 17.23
To find sin(101°15'), we convert degrees to radians:
101°15' = 101 + 15/60 = 101.25 degrees
101.25 degrees = (101.25 * π) / 180 radians
sin(101.25°) ≈ sin((101.25 * π) / 180)
Now we can calculate sin(A):
sin(A) ≈ (sin((101.25 * π) / 180) * a) / b
Since we don't know the value of a yet, we can't solve for A at this point.
Now, let's calculate the value of C using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C)
We are given:
b = 17.23cm
c = 10.86cm
Plugging in these values, we get:
10.86^2 = a^2 + 17.23^2 - 2 * a * 17.23 * cos(C)
Simplifying this equation will give us the value of cos(C). However, we still can't solve for C yet because we don't know the value of a.
To find the value of a, we can use the Law of Sines again:
sin(A) / a = sin(B) / b
Plugging in the values we know:
sin(A) / a = sin(101°15') / 17.23
Now we can solve for a:
a = (sin(101.25°) * 17.23) / sin(A)
To calculate sin(A), we can use the equation we derived earlier:
sin(A) ≈ (sin((101.25 * π) / 180) * a) / b
Finally, substitute this value of sin(A) in the equation for a:
a = (sin(101.25°) * 17.23) / [(sin((101.25 * π) / 180) * 17.23) / sin(A)]
Now that we know the value of a, we can substitute it back into the Law of Cosines equation and solve for cos(C):
10.86^2 = a^2 + 17.23^2 - 2 * a * 17.23 * cos(C)
Solve this equation for cos(C) and then use the inverse cosine function to find the value of C:
C = cos^(-1)[(a^2 + b^2 - c^2) / (2ab)]
Using these steps, you can solve for the values of a, A, and C in ∆ABC.
To calculate the value of a, A, and C in ∆Abc, we will use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. The formula for the Law of Sines is:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we are given the values for b, c, and B. We need to solve for a, A, and C. Let's go step by step:
1. Substitute the given values into the Law of Sines formula:
a/sin(A) = 17.23/sin(101°15') = 10.86/sin(C)
2. Rearrange the formula to solve for a:
a = sin(A) * (17.23/sin(101°15'))
3. Use a calculator to find the sine of the angle A:
sin(A) = a/(17.23/sin(101°15'))
4. Use the arcsin function (sin^-1) on your calculator to find the angle A:
A = arcsin(a/(17.23/sin(101°15')))
5. Calculate the value of C using the Law of Sines:
C = arcsin((10.86/sin(C))*(sin(A)))
Now, you can plug in the values of a, A, and C into the equations and solve for their respective values.