Sine gives the same value for an angle and its supplement. sin 30° and sin 150° are both 0.5, so an inverse sine hides a second possibility.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
When it matters
The trouble appears only with SSA: two sides and an angle that is not between them. Every other arrangement fixes the triangle.
Zero, one or two
The side opposite the known angle may be too short to reach, may reach at exactly one point, or may cross the far side twice.
The test
Find the angle, then check whether its supplement still leaves a positive third angle. If the two known angles already total 180° or more, the second triangle does not exist.
One way out
The Law of Cosines never has this problem, because cosine is negative for obtuse angles and positive for acute ones. The sign settles it.
Why SSA is ambiguous
Two sides and a non-included angle can sometimes be assembled into two different triangles, one acute and one obtuse. The given data genuinely does not determine the triangle, which is why SSA is not a congruence criterion.
No triangle, one, or two
Depending on the numbers there may be no solution, exactly one, or two. Checking which case applies before solving avoids reporting one answer when two exist, or one when none does.
Finding the second solution
The calculator returns the acute angle; the obtuse candidate is 180° minus it. Check whether that candidate leaves a positive third angle. If it does, both triangles are valid and both must be reported.
Only the sine rule is ambiguous
The cosine rule returns a unique angle because cosine is negative for obtuse angles and positive for acute ones, so its inverse distinguishes them. Sine cannot, which is precisely the source of the ambiguity.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0)
Step 3: Watch an Example
One step at a time.
Watch Diego Test for a Second Triangle
A triangle has a = 8, b = 10 and A = 40°.
- Step 1
The Law of Sines gives sin B = 10 × sin 40° ÷ 8, about 0.803.
Step 4: Your Turn
Practice makes it stick.
The Supplement
Problem 1 of 2
An inverse sine gives 35°. What is the supplement, in degrees?
degrees
The Count
Problem 2 of 2
A = 70° and the second candidate for B would be 130°. How many triangles are possible?
How Many Triangles?
1 of 8
An inverse sine gives 50°. The supplement, in degrees?
2 of 8
A = 30°, B could be 45° or 135°. How many triangles?
3 of 8
A = 100°, B could be 40° or 140°. How many triangles?
4 of 8
The Law of Sines gives sin B = 1.4. How many triangles?
5 of 8
A = 40°, B = 53.4°. What is the third angle, in degrees to one decimal place?
6 of 8
Which configurations can ever be ambiguous?
7 of 8
Put the ambiguous case check in order.
- 1Take the inverse sine for the first candidate.
- 2Subtract from 180° for the second candidate.
- 3Keep the second only if the two known angles stay under 180°.
- 4Use the Law of Sines to find sin B.
8 of 8
sin B = 0.5 and B must be obtuse. What is B, in degrees?
Step 5: Quick Check
Show what you know.
Question 1 of 2
An inverse sine gives 25°. What is the other candidate, in degrees?
Question 2 of 2
Why does the Law of Cosines avoid the ambiguity?
What You Learned
- An angle and its supplement share the same sine.
- Only SSA can give two triangles, one, or none.
- Keep the supplement only if the two known angles still total under 180°.