# Trigonometry : Ambiguous Triangles

## Example Questions

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### Example Question #1 : Ambiguous Triangles

Given  and  determine to the nearest degree the measure of .

Explanation:

We are given three sides and our desire is to find an angle, this means we must utilize the Law of Cosines. Since the angle desired is  the equation must be rewritten as such:

Substituting the given values:

Rearranging:

Solving the right hand side and taking the inverse cosine we obtain:

### Example Question #2 : Ambiguous Triangles

If  and , determine the measure of  to the nearest degree.

Explanation:

This is a straightforward Law of Cosines problem since we are given three sides and desire one of the corresponding angles in the triangle. We write down the Law of Cosines to start:

Substituting the given values:

Isolating the angle:

The final step is to take the inverse cosine of both sides:

### Example Question #3 : Ambiguous Triangles

If , and  determine the length of side , round to the nearest whole number.

Explanation:

This is a straightforward Law of Sines problem as we are given two angles and a corresponding side:

Substituting the known values:

Solving for the unknown side:

### Example Question #4 : Ambiguous Triangles

If , and  determine the measure of , round to the nearest degree.

Explanation:

This is a straightforward Law of Sines problem since we are given one angle and two sides and are asked to determine the corresponding angle.

Substituting the given values:

Now rearranging the equation:

The final step is to take the inverse sine of both sides:

### Example Question #5 : Ambiguous Triangles

If , , and  find  to the nearest degree.

Explanation:

The problem gives the lengths of three sides and asks to find an angle. We can use the Law of Cosines to solve for the angle. Because we are solving for , we use the equation:

Substituting the values from the problem gives

Isolating  by itself gives

### Example Question #6 : Ambiguous Triangles

If , find  to the nearest degree.

Explanation:

We are given the lengths of the three sides to a triangle. Therefore, we can use the Law of Cosines to find the angle being asked for. Since we are looking for  we use the equation,

Inputting the values we are given,

Next we isolate  by itself to solve for it

### Example Question #7 : Ambiguous Triangles

If , and , find  to the nearest degree.

Explanation:

Because the problem provides all three sides of the triangle, we can use the Law of Cosines to solve this problem. Since we are solving for , we use the equation

Substitute in the given values

Isolate

### Example Question #8 : Ambiguous Triangles

If  = , and  find  to the nearest degree.

Explanation:

Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. Because we are looking for  we use

Inputting the lengths of the triangle into this equation

Isolating

### Example Question #9 : Ambiguous Triangles

If , and  =  find  to the nearest degree.

Explanation:

Because we are given the length of two sides of a triangle and the corresponding angle of one of the sides, we can find the angle being asked for with the Law of Sines.

Inputting the values of the problem

Rearranging the equation to isolate

### Example Question #10 : Ambiguous Triangles

If  =  , , find  to the nearest tenth of a degree.