find values for b such that the triangle has no solution A=88 degree, a=315.6
assume angle B = 90°, then b has one solution, namely
sin88 = 315.6/b
b = 315.6/sin88 = 316.79
so there is no solution for b > 316.79
check: let b = 320
find angle B
sin88/315.6 = sinB/320
sinB = 320sin88/315.6 = 1.0133.. which is not possible.
To find values for b such that the triangle has no solution, we can 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.
In this case, we have the following information:
Angle A = 88 degrees
Side a = 315.6
Let's assign the remaining angle as angle B, and its corresponding side as b.
Using the Law of Sines:
sin(A) / a = sin(B) / b
Substituting the given values:
sin(88) / 315.6 = sin(B) / b
To determine if there is a solution, we need to find the maximum possible value for b such that sin(B) ≤ 1.
Since sin(B) cannot be greater than 1, we have:
sin(88) / 315.6 ≤ 1 / b
Rearranging the equation:
b ≤ 315.6 / sin(88)
Evaluating the right side:
b ≤ 315.6 / 0.99984766
b ≤ 315.634 (rounded to three decimal places)
Therefore, the values for b that satisfy the condition and do not yield a triangle solution are b ≤ 315.634.
To determine the values of b for which a triangle with the given measurements has no solution, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
In this case, we have an angle A of 88 degrees and a side a of length 315.6 units. As we want to find values for b, which is the side opposite to angle A, we will use the following formula:
b / sin(B) = a / sin(A)
First, we need to compute the sine of angle A:
sin(A) = sin(88°)
Using a calculator, we find that sin(88°) ≈ 0.9998.
Next, we can rearrange the formula to solve for sin(B):
sin(B) = (b * sin(A)) / a
Now, we can substitute the given values:
sin(B) = (b * 0.9998) / 315.6
For a triangle to have a solution, the sine of angle B must be less than or equal to 1. Thus, we can set up the following inequality:
sin(B) ≤ 1
Substituting the expression for sin(B) from above, we get:
(b * 0.9998) / 315.6 ≤ 1
To find the values of b that satisfy this inequality, we can solve it by multiplying both sides by 315.6:
b * 0.9998 ≤ 315.6
Next, divide both sides of the inequality by 0.9998 to isolate b:
b ≤ 315.6 / 0.9998
Calculating the right side of the inequality:
b ≤ 315.66
Therefore, the values for b that satisfy the inequality and ensure that the triangle has a solution are any values less than or equal to 315.66 units.