Question

An artist is designing triangular mirrors. Determine the number of different triangles that she can form using the given measurements. Then solve the triangles. Round to the nearest tenth.

a=4.2 cm b= 5.7 cm measure angle A= 39 degrees

Answer 1

• There are two solutions:
• B = 58.7°, C = 82.3°, c = 6.6 cm
• B = 121.3°, C = 19.7°, c = 2.2 cm

Explanation:

You are given a side and its opposite angle (a, A), so the Law of Sines can be used to solve the triangle. The side given is the shorter of the two given sides, so it is likely there are two solutions. (If the given side is the longer of the two, there will always be only one solution.)

The Law of Sines tells you ...

a/sin(A) = b/sin(B) = c/sin(C)

Of course, the sum of angles in a triangle is 180°, so once you find angle B, you can use that fact to find angle C, thus side c.

The solution process finds angle B first:

B = arcsin(b/a·sin(A)) . . . . . . or the supplement of this value

then angle C:

C = 180° -A -B = 141° -B

finally, side c:

c = a·sin(C)/sin(A)

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A triangle solver application for phone or tablet (or the one on your graphing calculator) can solve the triangle for you, or you can implement the above formulas in a spreadsheet (or even do them by hand). Of course, you need to pay attention to whether the functions involved give or take radians instead of degrees.