How To Find The Angle Of A Non Right Triangle

Finding the angles of a non-right triangle, also known as an oblique triangle, requires different approaches than right triangles. Instead of relying on the Pythagorean theorem and basic trigonometric ratios (SOH CAH TOA), we turn to the Law of Sines and the Law of Cosines. These laws provide the necessary relationships between sides and angles to solve for unknown angles.

Understanding Non-Right Triangles

Non-right triangles have no angle measuring 90 degrees. They come in two main varieties:

• Acute Triangles: All three angles are less than 90 degrees.
• Obtuse Triangles: One angle is greater than 90 degrees.

Solving for the angles in these triangles requires knowing at least three pieces of information:

• Three Sides (SSS)
• Two Sides and the Included Angle (SAS)
• Two Angles and a Side (AAS or ASA)

The Law of Sines

The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle. Mathematically:

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

Where:

• a, b, c are the lengths of the sides of the triangle.
• A, B, C are the angles opposite those sides, respectively.

When to Use the Law of Sines:

The Law of Sines is most useful when you have:

• AAS (Angle-Angle-Side): Two angles and a non-included side.
• ASA (Angle-Side-Angle): Two angles and the included side.
• SSA (Side-Side-Angle): Two sides and a non-included angle. This case can be tricky and may lead to the ambiguous case (more on that later).

Steps to Finding an Angle Using the Law of Sines:

1. Identify the Known Information: Determine which sides and angles you know. Make sure you have a side and its opposite angle known.
2. Set up the Proportion: Write out the Law of Sines, then plug in the known values. You'll have one ratio that's fully known and another with one unknown angle.
3. Solve for the Sine of the Unknown Angle: Cross-multiply and isolate the sine of the unknown angle.
4. Find the Angle Using the Inverse Sine Function: Use the inverse sine function (arcsin or sin^-1) on your calculator to find the measure of the angle. Remember that the sine function has a range of -1 to 1, so make sure the value you're taking the inverse sine of is within this range.
5. Consider the Ambiguous Case (SSA): If you used SSA, you might have two possible solutions. You'll need to check if both solutions are valid by seeing if they result in a triangle with angles that add up to 180 degrees.

Example: AAS

Suppose we have a triangle where:

• Angle A = 30 degrees
• Angle B = 70 degrees
• Side a = 8

We want to find angle C and side b.

First, find angle C:

Since the angles in a triangle add up to 180 degrees:

C = 180 - A - B = 180 - 30 - 70 = 80 degrees

Next, find side b using the Law of Sines:

a / sin(A) = b / sin(B)
8 / sin(30) = b / sin(70)

Cross-multiply:

8 * sin(70) = b * sin(30)

Isolate b:

b = (8 * sin(70)) / sin(30)
b ≈ (8 * 0.9397) / 0.5
b ≈ 15.035

Therefore, angle C is 80 degrees and side b is approximately 15.035.

Example: ASA

Suppose we have a triangle where:

• Angle A = 45 degrees
• Angle B = 60 degrees
• Side c = 12

We want to find angle C, side a and side b.

First, find angle C:

Since the angles in a triangle add up to 180 degrees:

C = 180 - A - B = 180 - 45 - 60 = 75 degrees

Next, find side a using the Law of Sines:

a / sin(A) = c / sin(C)
a / sin(45) = 12 / sin(75)

Cross-multiply:

a * sin(75) = 12 * sin(45)

Isolate a:

a = (12 * sin(45)) / sin(75)
a ≈ (12 * 0.7071) / 0.9659
a ≈ 8.77

Next, find side b using the Law of Sines:

b / sin(B) = c / sin(C)
b / sin(60) = 12 / sin(75)

Cross-multiply:

b * sin(75) = 12 * sin(60)

Isolate b:

b = (12 * sin(60)) / sin(75)
b ≈ (12 * 0.8660) / 0.9659
b ≈ 10.77

Therefore, angle C is 75 degrees, side a is approximately 8.77 and side b is approximately 10.77.

The Ambiguous Case (SSA)

The SSA case is called "ambiguous" because the given information can sometimes lead to two possible triangles, one triangle, or no triangle at all. This ambiguity arises because knowing two sides and a non-included angle doesn't uniquely define the triangle.

• No Triangle: If the side opposite the given angle is too short to reach the base, no triangle can be formed.
• One Triangle: If the side opposite the given angle is exactly the right length to reach the base at a right angle, or if it's longer than the adjacent side, there's only one possible triangle.
• Two Triangles: If the side opposite the given angle is shorter than the adjacent side but long enough to reach the base at two different points, two different triangles can be formed.

How to Handle the Ambiguous Case:

1. Solve for the First Possible Angle: Use the Law of Sines to find the first possible value for the unknown angle.
2. Find the Second Possible Angle: Subtract the first angle from 180 degrees to find the second possible angle (its supplement).
3. Check Validity: For each possible angle, add it to the given angle.
   • If the sum is less than 180 degrees, the angle is a valid solution, and a triangle can be formed.
   • If the sum is greater than or equal to 180 degrees, the angle is not a valid solution, and that triangle doesn't exist.
4. Solve for the Remaining Sides and Angles: If both angles are valid, you'll have two different triangles to solve, each with its own set of sides and angles.

Example: SSA

Suppose we have a triangle where:

• Side a = 20
• Side b = 15
• Angle A = 40 degrees

We want to find angles B, C, and side c.

1. Solve for the First Possible Angle B:

a / sin(A) = b / sin(B)
20 / sin(40) = 15 / sin(B)
sin(B) = (15 * sin(40)) / 20
sin(B) ≈ (15 * 0.6428) / 20
sin(B) ≈ 0.4821
B ≈ arcsin(0.4821)
B ≈ 28.82 degrees

2. Find the Second Possible Angle B:

B' = 180 - 28.82
B' ≈ 151.18 degrees

3. Check Validity:

   • For B ≈ 28.82 degrees: A + B = 40 + 28.82 = 68.82 degrees. This is less than 180, so it's a valid solution.
   • For B' ≈ 151.18 degrees: A + B' = 40 + 151.18 = 191.18 degrees. This is greater than 180, so it's not a valid solution. Only the first solution for angle B is valid.

4. Solve for the Remaining Angle and Side (using B ≈ 28.82 degrees):

   • C = 180 - A - B = 180 - 40 - 28.82 = 111.18 degrees
   • Using the Law of Sines again to find side c:

a / sin(A) = c / sin(C)
20 / sin(40) = c / sin(111.18)
c = (20 * sin(111.18)) / sin(40)
c ≈ (20 * 0.9325) / 0.6428
c ≈ 29.01

In this case, only one triangle is possible: A = 40 degrees, B ≈ 28.82 degrees, C ≈ 111.18 degrees, a = 20, b = 15, and c ≈ 29.01.

The Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. There are three forms, each isolating a different angle:

• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)

Where:

• a, b, c are the lengths of the sides of the triangle.
• A, B, C are the angles opposite those sides, respectively.

When to Use the Law of Cosines:

The Law of Cosines is most useful when you have:

• SSS (Side-Side-Side): Three sides of the triangle are known.
• SAS (Side-Angle-Side): Two sides and the included angle are known.

Steps to Finding an Angle Using the Law of Cosines:

1. Identify the Known Information: Determine which sides and angles you know.
2. Choose the Correct Formula: Select the form of the Law of Cosines that isolates the angle you want to find.
3. Plug in the Values: Substitute the known side lengths into the formula.
4. Solve for the Cosine of the Unknown Angle: Simplify the equation and isolate the cosine of the unknown angle.
5. Find the Angle Using the Inverse Cosine Function: Use the inverse cosine function (arccos or cos^-1) on your calculator to find the measure of the angle.

Example: SSS

Suppose we have a triangle where:

• a = 5
• b = 7
• c = 8

We want to find all three angles.

First, let's find angle A:

Using the Law of Cosines:

a² = b² + c² - 2bc * cos(A)
5² = 7² + 8² - 2 * 7 * 8 * cos(A)
25 = 49 + 64 - 112 * cos(A)
25 = 113 - 112 * cos(A)
-88 = -112 * cos(A)
cos(A) = -88 / -112
cos(A) = 0.7857
A = arccos(0.7857)
A ≈ 38.21 degrees

Next, let's find angle B:

Using the Law of Cosines:

b² = a² + c² - 2ac * cos(B)
7² = 5² + 8² - 2 * 5 * 8 * cos(B)
49 = 25 + 64 - 80 * cos(B)
49 = 89 - 80 * cos(B)
-40 = -80 * cos(B)
cos(B) = -40 / -80
cos(B) = 0.5
B = arccos(0.5)
B = 60 degrees

Finally, find angle C:

Since the angles in a triangle add up to 180 degrees:

C = 180 - A - B = 180 - 38.21 - 60 = 81.79 degrees

Therefore, A ≈ 38.21 degrees, B = 60 degrees, and C ≈ 81.79 degrees.

Example: SAS

Suppose we have a triangle where:

• a = 10
• b = 12
• Angle C = 50 degrees

We want to find side c and angles A and B.

First, find side c using the Law of Cosines:

c² = a² + b² - 2ab * cos(C)
c² = 10² + 12² - 2 * 10 * 12 * cos(50)
c² = 100 + 144 - 240 * cos(50)
c² = 244 - 240 * 0.6428
c² = 244 - 154.27
c² = 89.73
c = √89.73
c ≈ 9.47

Next, find angle A using the Law of Sines (or the Law of Cosines – either works here):

a / sin(A) = c / sin(C)
10 / sin(A) = 9.47 / sin(50)
sin(A) = (10 * sin(50)) / 9.47
sin(A) ≈ (10 * 0.7660) / 9.47
sin(A) ≈ 0.8089
A = arcsin(0.8089)
A ≈ 54.02 degrees

Finally, find angle B:

Since the angles in a triangle add up to 180 degrees:

B = 180 - A - C = 180 - 54.02 - 50 = 75.98 degrees

Therefore, c ≈ 9.47, A ≈ 54.02 degrees, and B ≈ 75.98 degrees.

Choosing Between the Law of Sines and the Law of Cosines

Here's a summary to help you decide which law to use:

• Law of Sines: Use when you have AAS, ASA, or SSA. Be particularly careful with SSA due to the ambiguous case.
• Law of Cosines: Use when you have SSS or SAS.

In some cases, you can use either law after finding some initial information. For example, if you start with SAS and use the Law of Cosines to find the third side, you can then use the Law of Sines to find the remaining angles (although using the Law of Cosines again for the angles can avoid potential ambiguity issues).

Practical Applications

Understanding how to find angles in non-right triangles has numerous practical applications in fields like:

• Surveying: Determining distances and elevations in land surveying.
• Navigation: Calculating courses and distances in air and sea navigation.
• Engineering: Designing structures, bridges, and other civil engineering projects.
• Physics: Analyzing forces and motion in mechanics.
• Astronomy: Calculating distances and angles between celestial objects.

Common Mistakes and How to Avoid Them

• Incorrectly Identifying Given Information: Make sure you correctly identify which sides and angles are given and their relationships (opposite, included).
• Using the Wrong Law: Choose the appropriate law based on the given information (SSS, SAS, AAS, ASA, SSA).
• Ignoring the Ambiguous Case (SSA): Always check for two possible solutions when using the Law of Sines with SSA.
• Calculator Errors: Ensure your calculator is in degree mode, not radian mode. Double-check your inputs, especially when using inverse trigonometric functions.
• Rounding Errors: Avoid rounding intermediate calculations too early. Round only the final answer to the desired level of precision.
• Forgetting the Angle Sum Property: The angles in any triangle must add up to 180 degrees. This is a useful check for your calculations.

Conclusion

Finding the angles of a non-right triangle involves using the Law of Sines and the Law of Cosines, understanding their applications, and being aware of the potential ambiguous case. By carefully applying these laws and paying attention to detail, you can accurately solve for unknown angles in various real-world scenarios. Mastering these techniques provides a solid foundation for more advanced trigonometry and its applications in diverse fields. Remember to practice regularly and pay attention to the details to avoid common mistakes.