Question

Use the Law of Sines to find all triangles if β=115∘,b=18.25,c=15.25. Round to two decimal places. As in the text, (α,a),(β,b) and (γ,c) are angle-side opposite pairs. If no such triangle exists, enter DNE in each answer box. Enter the values for an acute triangle solution:

γ= __________ degrees
α= _________ degrees
a=​ _________
Enter the values for an obtuse triangle solution:
γ′= _______ degrees
α′= ________ degrees
a′=​ ________

Answer 1

To find all possible triangles with the given values using the Law of Sines, one must calculate the angle γ by relating it with angle β and side c. After finding γ, determine α and side a. Remember, only one valid triangle exists since β is already 115° which constrains γ to be less than 65°.

Step-by-step explanation:

To solve this problem, we need to use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is the same for all sides and angles in a triangle.

For a triangle with sides a, b, and c, and angles α (alpha), β (beta), and γ (gamma), the Law of Sines is written as:

a b c
--------- = --------- = ---------
sin(α) sin(β) sin(γ)

We have β = 115°, b = 18.25, and c = 15.25. We first find angle γ using the Law of Sines:

15.25 18.25
------------ = ------------
sin(γ) sin(115°)