Trigonometry On Non Right Angled Triangles

Trigonometry extends its reach beyond the familiar right-angled triangles, offering powerful tools to analyze any triangle, regardless of its angles. The sine rule and cosine rule are the cornerstones of this extension, allowing us to solve for unknown sides and angles in oblique (non-right angled) triangles. This article delves into these rules, their applications, and nuances.

The Sine Rule: Unveiling Relationships Between Sides and Angles

The sine rule establishes a proportional relationship between the sides of a triangle and the sines of their opposite angles. In any triangle ABC, with sides a, b, c opposite to angles A, B, and C respectively, the sine rule states:

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

This rule provides a powerful tool for solving triangles when we know:

• Two angles and one side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA - the ambiguous case)

Applications of the Sine Rule

Let's explore how the sine rule is applied in different scenarios:

1. Finding an Unknown Side (AAS or ASA):

Imagine a triangle where angle A = 40°, angle B = 60°, and side a = 10 cm. To find side b, we use the sine rule:

10 / sin(40°) = b / sin(60°)

Solving for b:

b = 10 * sin(60°) / sin(40°) ≈ 13.47 cm

2. Finding an Unknown Angle (AAS or ASA):

Consider a triangle where angle A = 30°, side a = 8 cm, and side b = 12 cm. To find angle B:

8 / sin(30°) = 12 / sin(B)

Solving for sin(B):

sin(B) = 12 * sin(30°) / 8 = 0.75

Therefore, angle B = arcsin(0.75) ≈ 48.59°. Remember to consider the ambiguous case discussed later!

The Ambiguous Case (SSA)

The SSA case, where we know two sides and a non-included angle, is known as the ambiguous case because it can potentially lead to zero, one, or two possible triangle solutions. This ambiguity arises because the given information might allow for different triangle configurations.

Understanding the Possibilities:

Let's say we have triangle ABC with known side a, side b, and angle A. We want to find angle B.

• No Solution: If a is too short relative to b and angle A, it won't reach the base to form a triangle. Mathematically, this occurs when a < bsin(A)*. In this case, sin(B) would be greater than 1, which is impossible.

• One Solution:

  • If a = bsin(A)*, then a is exactly the right length to form a right-angled triangle. Angle B = 90°.
  • If a >= b, then a is long enough to reach the base in only one way, forming a single triangle.

• Two Solutions: If bsin(A) < a < b, then a is long enough to reach the base, but it can do so in two different locations, creating two possible triangles.

Resolving the Ambiguity:

To determine the correct number of solutions and find the angles, follow these steps:

1. Use the Sine Rule: Calculate sin(B) using the sine rule: sin(B) = bsin(A) / a.
2. Check for Validity:
   • If sin(B) > 1, there is no solution.
   • If sin(B) = 1, then B = 90°, and there is one solution.
   • If sin(B) < 1, find the principal value of B: B1 = arcsin(sin(B)).
3. Consider the Second Possible Angle: Calculate the supplementary angle: B2 = 180° - B1.
4. Check if the Second Angle is Valid: Determine if B2 is a valid angle by checking if A + B2 < 180°.
   • If A + B2 < 180°, then there are two possible triangles.
   • If A + B2 >= 180°, then there is only one possible triangle (using angle B1).
5. Solve for the Remaining Angles and Sides: Once you have the possible values for angle B, you can calculate the remaining angles and sides for each possible triangle using the sine rule and the fact that the angles in a triangle sum to 180°.

Example:

Let's say a = 5 cm, b = 8 cm, and A = 30°.

1. sin(B) = (8 * sin(30°)) / 5 = 0.8
2. Since sin(B) < 1, we proceed.
3. B1 = arcsin(0.8) ≈ 53.13°
4. B2 = 180° - 53.13° ≈ 126.87°
5. Check validity of B2: 30° + 126.87° = 156.87° < 180°. Therefore, there are two possible triangles.

Now, we solve for each triangle separately:

• Triangle 1: A = 30°, B = 53.13°, C = 180° - 30° - 53.13° = 96.87°. Use the sine rule to find side c.
• Triangle 2: A = 30°, B = 126.87°, C = 180° - 30° - 126.87° = 23.13°. Use the sine rule to find side c.

The Cosine Rule: Relating Sides and Angles Through a Different Lens

The cosine rule provides a relationship between the sides of a triangle and the cosine of one of its angles. It's particularly useful when the sine rule falls short. For triangle ABC:

• a² = b² + c² - 2bccos(A)*
• b² = a² + c² - 2accos(B)*
• c² = a² + b² - 2abcos(C)*

The cosine rule is invaluable when we know:

• Three sides (SSS)
• Two sides and the included angle (SAS)

Applications of the Cosine Rule

Let's examine the applications of the cosine rule:

1. Finding an Unknown Side (SAS):

Suppose we know b = 7 cm, c = 9 cm, and angle A = 50°. To find side a:

a² = 7² + 9² - 2 * 7 * 9 * cos(50°) a² ≈ 49 + 81 - 79.97 ≈ 50.03 a ≈ √50.03 ≈ 7.07 cm

2. Finding an Unknown Angle (SSS):

Given a = 5 cm, b = 8 cm, and c = 7 cm, let's find angle A. We rearrange the cosine rule:

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

When to Use Sine Rule vs. Cosine Rule: A Practical Guide

Choosing between the sine and cosine rules depends on the information provided:

• Sine Rule: Use when you have a side and its opposite angle, along with one other piece of information (either another angle or another side). This is best for AAS, ASA, and SSA (remembering the ambiguous case!).
• Cosine Rule: Use when you have three sides (SSS) or two sides and the included angle (SAS).

In summary:

• AAS, ASA, SSA (ambiguous): Sine Rule
• SSS, SAS: Cosine Rule

Beyond Solving Triangles: Applications in Real-World Scenarios

Trigonometry on non-right angled triangles extends far beyond academic exercises. It finds application in numerous real-world scenarios:

• Navigation: Calculating distances and bearings in air and sea navigation relies heavily on the sine and cosine rules, especially when dealing with non-right angled courses.
• Surveying: Surveyors use these rules to determine distances and angles on land, particularly in areas with uneven terrain.
• Engineering: Engineers apply trigonometric principles to calculate forces, stresses, and strains in structures like bridges and buildings. They also use it in designing mechanisms and analyzing motion.
• Astronomy: Astronomers utilize trigonometry to measure distances to stars and planets, analyze their movements, and understand celestial relationships.
• Cartography: Creating accurate maps requires precise measurements of distances and angles, which are facilitated by trigonometric calculations.
• Computer Graphics: Trigonometry is fundamental to computer graphics, allowing for the creation of realistic 3D models, animations, and game environments. Calculating angles of reflection and refraction of light is crucial for realistic rendering.

Advanced Concepts and Considerations

While the sine and cosine rules are fundamental, a deeper understanding of trigonometry involves exploring related concepts:

• Area of a Triangle: Given two sides and the included angle, the area of a triangle can be calculated using the formula: Area = (1/2) * b * c * sin(A). This formula is directly derived from trigonometric principles.
• Heron's Formula: If all three sides of a triangle are known, Heron's formula provides an alternative method for calculating the area: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter: s = (a + b + c) / 2.
• Vector Analysis: Trigonometry is closely related to vector analysis. Vectors can be resolved into components using trigonometric functions, which is essential for analyzing forces, velocities, and other vector quantities.
• Spherical Trigonometry: For triangles drawn on the surface of a sphere (important in navigation and astronomy on a global scale), standard planar trigonometry does not apply. Spherical trigonometry provides specialized rules and formulas for dealing with these curved triangles.

Common Pitfalls and How to Avoid Them

Applying the sine and cosine rules effectively requires careful attention to detail. Here are some common mistakes and how to avoid them:

• Ambiguous Case Errors: Failing to recognize and properly address the ambiguous case (SSA) is a frequent source of error. Always check for the possibility of two solutions.
• Incorrect Angle-Side Pairing: Ensure you are using the correct angle-side pairings in the sine rule. The side a must be opposite angle A, and so on.
• Calculator Settings: Verify that your calculator is in the correct angle mode (degrees or radians) before performing trigonometric calculations.
• Rounding Errors: Avoid rounding intermediate results, as this can lead to significant errors in the final answer. Keep as many decimal places as possible until the final step.
• Misinterpreting the Cosine Rule Formula: Ensure you are using the correct form of the cosine rule formula, depending on which side or angle you are trying to find. Pay close attention to the signs and the placement of the sides.
• Forgetting Units: Always include the correct units in your final answer (e.g., cm, meters, degrees).

Example Problems with Detailed Solutions

Let's solidify our understanding with some example problems:

Problem 1:

In triangle PQR, PQ = 12 cm, PR = 8 cm, and angle P = 70°. Find the length of QR.

Solution:

We have two sides and the included angle (SAS), so we use the cosine rule:

QR² = PQ² + PR² - 2 * PQ * PR * cos(P) QR² = 12² + 8² - 2 * 12 * 8 * cos(70°) QR² = 144 + 64 - 192 * cos(70°) QR² ≈ 208 - 192 * 0.3420 QR² ≈ 208 - 65.66 QR² ≈ 142.34 QR ≈ √142.34 ≈ 11.93 cm

Therefore, QR ≈ 11.93 cm.

Problem 2:

In triangle ABC, AB = 10 cm, BC = 7 cm, and angle A = 35°. Find the possible values of angle C.

Solution:

We have two sides and a non-included angle (SSA), so we must consider the ambiguous case. We use the sine rule:

10 / sin(C) = 7 / sin(35°) sin(C) = (10 * sin(35°)) / 7 sin(C) ≈ (10 * 0.5736) / 7 ≈ 0.8194

Now, we find the principal value of C:

C1 = arcsin(0.8194) ≈ 55.03°

Next, we find the supplementary angle:

C2 = 180° - 55.03° ≈ 124.97°

Now, we check if C2 is a valid angle:

A + C2 = 35° + 124.97° = 159.97° < 180°

Since A + C2 < 180°, there are two possible triangles.

Therefore, the possible values of angle C are approximately 55.03° and 124.97°.

Problem 3:

The sides of a triangle are 4 cm, 5 cm, and 6 cm. Find the largest angle.

Solution:

The largest angle is opposite the longest side. Let a = 4 cm, b = 5 cm, and c = 6 cm. We want to find angle C.

We use the cosine rule:

c² = a² + b² - 2abcos(C) 6² = 4² + 5² - 2 * 4 * 5 * cos(C) 36 = 16 + 25 - 40cos(C) 36 = 41 - 40cos(C) 40cos(C) = 41 - 36 = 5 cos(C) = 5 / 40 = 0.125 C = arccos(0.125) ≈ 82.82°

Therefore, the largest angle is approximately 82.82°.

Conclusion: Mastering the Triangle

The sine and cosine rules provide powerful tools for analyzing and solving non-right angled triangles. By understanding their applications, limitations, and potential pitfalls, you can confidently tackle a wide range of trigonometric problems, from academic exercises to real-world applications in navigation, surveying, engineering, and beyond. Mastering these rules unlocks a deeper understanding of geometric relationships and their practical significance. Remember to practice applying these concepts to various problems to solidify your understanding and develop your problem-solving skills.