Find the indicated angle è.
(Use either the Law of Sines or the Law of Cosines, as appropriate. Assume a = 110 and c = 136 (angle B=38)
Round your answer to two decimal places.)
Law of Cosines:
cosB = (a^2+c^2-b^2)/2ac.
cos38 = ((110)^2+(136)^2-b^2) / 29920.
0.7880 = (30596-b^2) / 29920.
Cross multiply:
30,596-b^2 = 0.7880*29920 = 23576.96.
-b^2 = 23,976.96-30596 = -6619.04.
b^2 = 6619.04.
b = 81.36.
Law of Sines:
sinA/a = sinB/b.
sinA / 110 = sin38 / 81.36.
Multiply both sides by 110:
sinA = 110(sin38 / 81.36) = 0.832384.
A = 56.34 Deg.
C = 180 - 38 - 56.34 = 85.66 Deg.
To find angle A (indicated as è), we can use the Law of Cosines.
The Law of Cosines states that:
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, and C is the angle opposite side c.
In this case, we are given side a = 110, side c = 136, and angle B = 38.
We can rearrange the formula to solve for the angle C:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Plugging in the known values, we get:
cos(C) = (110^2 + b^2 - 136^2) / (2 * 110 * b)
Simplifying the equation gives:
cos(C) = (12100 + b^2 - 18496) / (220b)
cos(C) = (b^2 - 6396) / (220b)
Now, since we know that the sum of all angles in a triangle is 180 degrees, we can conclude that:
A + B + C = 180
Since we are given angle B = 38 degrees, we can substitute this into the equation:
A + 38 + C = 180
Rearranging the equation to find angle A:
A = 180 - 38 - C
A = 142 - C
Now, we have two equations:
cos(C) = (b^2 - 6396) / (220b)
A = 142 - C
We can substitute the value of A into the equation cos(C) = (b^2 - 6396) / (220b):
cos(C) = (b^2 - 6396) / (220b)
cos(142 - C) = (b^2 - 6396) / (220b)
Now, we can solve for angle C using trigonometric functions:
C = 142 - arccos((b^2 - 6396) / (220b))
Since we are not given the value of side b, we cannot solve for angle C directly. Additional information is required to find the value of angle C.
To find the indicated angle è, we can use the Law of Sines since we have the measurements of one side and its opposite angle. 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 the same for all sides and angles.
The formula for the Law of Sines is:
sin(A)/a = sin(B)/b = sin(C)/c
Given that angle A is the indicated angle, side a is the side opposite angle A:
sin(A)/110 = sin(38)/b
To find angle A, we need to isolate sin(A) by cross-multiplying:
sin(A) = (110 * sin(38))/b
Now, we can solve for sin(A) by dividing both sides of the equation by b:
sin(A) = (110 * sin(38))/b
To find angle A, we can take the inverse sine (also known as arcsine) of both sides of the equation:
A = arcsin((110 * sin(38))/b)
Substituting the given values a = 110, c = 136, and B = 38:
A = arcsin((110 * sin(38))/136)
Now we can calculate the value of A using a calculator or an online tool. After rounding to two decimal places, the value of angle A, è, will be obtained.