How To Solve Non Right Triangles

Solving non-right triangles might seem daunting at first, but with the right tools and understanding, it becomes a manageable task. The key lies in leveraging the Law of Sines and the Law of Cosines, which provide the necessary relationships between angles and sides in any triangle.

Understanding Non-Right Triangles

A non-right triangle, also known as an oblique triangle, is any triangle that does not contain a 90-degree angle. These triangles can be further classified into:

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

The absence of a right angle means we can't directly apply the familiar trigonometric ratios like sine, cosine, and tangent in the same way we do with right triangles. Instead, we rely on the Law of Sines and the Law of Cosines.

Essential Tools: Law of Sines and Law of Cosines

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, it's expressed as:

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

Where:

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

When to use the Law of Sines:

The Law of Sines is particularly useful when you know:

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 sometimes lead to the ambiguous case, which we'll discuss later).

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It's essentially a generalization of the Pythagorean theorem. The Law of Cosines can be expressed in three different forms:

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

Where:

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

When to use the Law of Cosines:

The Law of Cosines is most useful when you know:

SSS (Side-Side-Side): All three sides of the triangle.
SAS (Side-Angle-Side): Two sides and the included angle.

Step-by-Step Guide to Solving Non-Right Triangles

Here's a detailed guide on how to solve non-right triangles, broken down by the information you're given:

Case 1: AAS (Angle-Angle-Side)

Scenario: You are given two angles and a non-included side.

Steps:

Find the third angle: Since the sum of the angles in a triangle is always 180 degrees, you can easily find the third angle by subtracting the two known angles from 180 degrees.
```
C = 180° - A - B
```
Apply the Law of Sines: Use the Law of Sines to find the remaining sides. Choose the ratio that includes the known side and its opposite angle, then set it equal to the ratio involving one of the unknown sides and its opposite angle.
```
a / sin(A) = b / sin(B) = c / sin(C)
```
For example, if you know a, A, and B, you can find b as follows:
```
b = (a * sin(B)) / sin(A)
```
Repeat this process to find the remaining side.

Example:

Let's say you have a triangle where:

A = 30°
B = 70°
a = 8 cm

Solution:

Find angle C:
```
C = 180° - 30° - 70° = 80°
```

Find side b:

b = (8 * sin(70°)) / sin(30°)
b ≈ (8 * 0.9397) / 0.5
b ≈ 15.03 cm

Find side c:

c = (8 * sin(80°)) / sin(30°)
c ≈ (8 * 0.9848) / 0.5
c ≈ 15.76 cm

Case 2: ASA (Angle-Side-Angle)

Scenario: You are given two angles and the included side.

Steps:

Find the third angle: Similar to the AAS case, find the third angle by subtracting the two known angles from 180 degrees.
```
C = 180° - A - B
```
Apply the Law of Sines: Use the Law of Sines to find the remaining sides. You now have one side and its opposite angle, allowing you to easily set up the ratios.

Example:

Let's say you have a triangle where:

A = 40°
B = 60°
c = 12 m

Solution:

Find angle C:
```
C = 180° - 40° - 60° = 80°
```

Find side a:

a = (12 * sin(40°)) / sin(80°)
a ≈ (12 * 0.6428) / 0.9848
a ≈ 7.83 m

Find side b:

b = (12 * sin(60°)) / sin(80°)
b ≈ (12 * 0.8660) / 0.9848
b ≈ 10.57 m

Case 3: SSA (Side-Side-Angle) - The Ambiguous Case

Scenario: You are given two sides and a non-included angle. This is known as the ambiguous case because it can lead to zero, one, or two possible solutions.

Steps:

Apply the Law of Sines: Set up the Law of Sines to find the sine of the angle opposite one of the known sides.
Find the possible angles: Solve for the angle using the inverse sine function (arcsin or sin⁻¹). Remember that the sine function has two possible solutions between 0° and 180° because sin(θ) = sin(180° - θ).
Check for valid solutions:
- Solution 1: Use the angle found in step 2. Add it to the given angle. If the sum is less than 180°, then this is a valid solution. Find the third angle by subtracting the sum from 180°, and then use the Law of Sines to find the remaining side.
- Solution 2: Use (180° - the angle found in step 2). Add it to the given angle. If the sum is less than 180°, then this is also a valid solution. Find the third angle by subtracting the sum from 180°, and then use the Law of Sines to find the remaining side.
- If neither solution is valid (i.e., the sum of the angles is greater than 180°), then there is no triangle that satisfies the given conditions.
- If only one solution is valid, then there is only one possible triangle.

Example 1: No Solution

Let's say you have a triangle where:

a = 5 cm
b = 12 cm
A = 30°

Solution:

Apply the Law of Sines:

sin(B) / b = sin(A) / a
sin(B) = (b * sin(A)) / a
sin(B) = (12 * sin(30°)) / 5
sin(B) = (12 * 0.5) / 5
sin(B) = 1.2

Find the possible angles: Since the sine function cannot be greater than 1, there is no solution for angle B. Therefore, no triangle can be formed with these dimensions.

Example 2: One Solution

Let's say you have a triangle where:

a = 22 cm
b = 12 cm
A = 42°

Solution:

Apply the Law of Sines:

sin(B) / b = sin(A) / a
sin(B) = (b * sin(A)) / a
sin(B) = (12 * sin(42°)) / 22
sin(B) ≈ (12 * 0.6691) / 22
sin(B) ≈ 0.3651

Find the possible angles:

B = arcsin(0.3651)
B ≈ 21.41°

The other possible angle is:

180° - 21.41° = 158.59°

Check for valid solutions:

Solution 1:

A + B = 42° + 21.41° = 63.41° < 180°  (Valid)
C = 180° - 63.41° = 116.59°

Now, find side c:

c = (a * sin(C)) / sin(A)
c = (22 * sin(116.59°)) / sin(42°)
c ≈ (22 * 0.8942) / 0.6691
c ≈ 29.39 cm

Solution 2:

A + B = 42° + 158.59° = 200.59° > 180° (Invalid)

Therefore, there is only one valid solution:

B ≈ 21.41°
C ≈ 116.59°
c ≈ 29.39 cm

Example 3: Two Solutions

Let's say you have a triangle where:

a = 15 cm
b = 20 cm
A = 35°

Solution:

Apply the Law of Sines:

sin(B) / b = sin(A) / a
sin(B) = (b * sin(A)) / a
sin(B) = (20 * sin(35°)) / 15
sin(B) ≈ (20 * 0.5736) / 15
sin(B) ≈ 0.7648

Find the possible angles:

B = arcsin(0.7648)
B ≈ 49.93°

The other possible angle is:

180° - 49.93° = 130.07°

Check for valid solutions:

Solution 1:

A + B = 35° + 49.93° = 84.93° < 180° (Valid)
C = 180° - 84.93° = 95.07°

Now, find side c:

c = (a * sin(C)) / sin(A)
c = (15 * sin(95.07°)) / sin(35°)
c ≈ (15 * 0.9961) / 0.5736
c ≈ 26.04 cm

Solution 2:

A + B = 35° + 130.07° = 165.07° < 180° (Valid)
C = 180° - 165.07° = 14.93°

Now, find side c:

c = (a * sin(C)) / sin(A)
c = (15 * sin(14.93°)) / sin(35°)
c ≈ (15 * 0.2576) / 0.5736
c ≈ 6.74 cm

Therefore, there are two valid solutions:

Solution 1:

B ≈ 49.93°
C ≈ 95.07°
c ≈ 26.04 cm

Solution 2:

B ≈ 130.07°
C ≈ 14.93°
c ≈ 6.74 cm

Case 4: SSS (Side-Side-Side)

Scenario: You are given all three sides of the triangle.

Steps:

Apply the Law of Cosines: Use the Law of Cosines to find one of the angles. It's generally a good practice to find the largest angle first (the angle opposite the longest side) to avoid potential issues with the inverse cosine function.
```
cos(A) = (b² + c² - a²) / (2bc)
cos(B) = (a² + c² - b²) / (2ac)
cos(C) = (a² + b² - c²) / (2ab)
```
Solve for the angle: Use the inverse cosine function (arccos or cos⁻¹) to find the angle.
Apply the Law of Sines or Law of Cosines: You can now use either the Law of Sines or the Law of Cosines to find the remaining angles. The Law of Sines is often easier to use at this point.

Example:

Let's say you have a triangle where:

a = 5 cm
b = 7 cm
c = 8 cm

Solution:

Find the largest angle (C):

cos(C) = (a² + b² - c²) / (2ab)
cos(C) = (5² + 7² - 8²) / (2 * 5 * 7)
cos(C) = (25 + 49 - 64) / 70
cos(C) = 10 / 70
cos(C) = 1 / 7

Solve for angle C:
```
C = arccos(1/7)
C ≈ 81.79°
```

Find angle A using the Law of Sines:

sin(A) / a = sin(C) / c
sin(A) = (a * sin(C)) / c
sin(A) = (5 * sin(81.79°)) / 8
sin(A) ≈ (5 * 0.9898) / 8
sin(A) ≈ 0.6186

Solve for angle A:
```
A = arcsin(0.6186)
A ≈ 38.20°
```

Find angle B:

B = 180° - A - C
B = 180° - 38.20° - 81.79°
B ≈ 60.01°

Case 5: SAS (Side-Angle-Side)

Scenario: You are given two sides and the included angle.

Steps:

Apply the Law of Cosines: Use the Law of Cosines to find the side opposite the given angle.

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

Solve for the side: Take the square root of both sides to find the length of the unknown side.
Apply the Law of Sines or Law of Cosines: You can now use either the Law of Sines or the Law of Cosines to find the remaining angles. The Law of Sines is often easier to use at this point.

Example:

Let's say you have a triangle where:

a = 10 cm
b = 15 cm
C = 30°

Solution:

Find side c using the Law of Cosines:

c² = a² + b² - 2ab * cos(C)
c² = 10² + 15² - 2 * 10 * 15 * cos(30°)
c² = 100 + 225 - 300 * (√3 / 2)
c² = 325 - 300 * 0.8660
c² = 325 - 259.8
c² ≈ 65.2

Solve for side c:
```
c = √65.2
c ≈ 8.07 cm
```

Find angle A using the Law of Sines:

sin(A) / a = sin(C) / c
sin(A) = (a * sin(C)) / c
sin(A) = (10 * sin(30°)) / 8.07
sin(A) ≈ (10 * 0.5) / 8.07
sin(A) ≈ 0.6196

Solve for angle A:
```
A = arcsin(0.6196)
A ≈ 38.30°
```

Find angle B:

B = 180° - A - C
B = 180° - 38.30° - 30°
B ≈ 111.70°

Practical Tips and Considerations

Drawing a diagram: Always start by drawing a clear diagram of the triangle. Label the sides and angles with the given information. This helps visualize the problem and identify which law to apply.
Using a calculator: Make sure you are comfortable using your calculator to find sine, cosine, and inverse trigonometric functions. Be mindful of whether your calculator is in degree or radian mode.
Rounding: Avoid rounding intermediate calculations to maintain accuracy. Round your final answers to an appropriate number of significant figures.
Checking for reasonableness: After finding the unknown sides and angles, check if the results make sense. For example, the largest angle should be opposite the longest side, and the sum of the angles should be 180 degrees.
The ambiguous case (SSA): Be particularly careful when dealing with the SSA case. Always check for the possibility of two solutions.

Conclusion

Solving non-right triangles requires a solid understanding of the Law of Sines and the Law of Cosines, as well as careful attention to detail. By identifying the given information, choosing the appropriate law, and following the steps outlined above, you can successfully find the missing sides and angles of any oblique triangle. Remember to be mindful of the ambiguous case (SSA) and always check for reasonableness in your answers. With practice, you'll become proficient in navigating the world of non-right triangles.

How To Solve Non Right Triangles

Table of Contents

Understanding Non-Right Triangles

Essential Tools: Law of Sines and Law of Cosines

Law of Sines

Law of Cosines

Step-by-Step Guide to Solving Non-Right Triangles

Case 1: AAS (Angle-Angle-Side)

Case 2: ASA (Angle-Side-Angle)

Case 3: SSA (Side-Side-Angle) - The Ambiguous Case

Case 4: SSS (Side-Side-Side)

Case 5: SAS (Side-Angle-Side)

Practical Tips and Considerations

Conclusion

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