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Mastering the Law of Sines: Your Key to Solving Triangle Problems
Unlock the power of the Law of Sines to solve complex triangle problems. Learn how to apply this fundamental trigonometric principle to real-world scenarios in navigation, surveying, and more.

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Now Playing:Law of sines– Example 0
Intros
  1. The Law of Sines
Examples
  1. Given the following triangle △ABC\triangle ABC,
    Using law of sines to find angles and side lengths of triangles
    1. Solve for∠C\angle C

    2. Solve for aa

Angle in standard position
Jump to:NotesConceptExampleFAQsPrerequisitesRelated
Notes
In this section, we will learn about the Law of Sines, also known as the Sines Rule. The Law of Sines is a formula that models the relationship between the sides and the angles of any triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle. In order to use the Law of Sines, we need to satisfy the "one pair, one additional information" condition (i.e. Angle-Angle-Side abbreviated as AAS, and Angle-Side-Angle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines.
Law of Sine
For any △\triangle ABC,

asin⁡(A)\frac{a}{\sin(A)} =bsin⁡(B)=\frac{b}{\sin(B)} =csin⁡(C)=\frac{c}{\sin(C)}
and,
sin⁡(A)a\frac{\sin(A)}{a} =sin⁡(B)b=\frac{\sin(B)}{b} =sin⁡(C)c=\frac{\sin(C)}{c}

Use the Law of Sine when given a pair!

Ambiguous case
Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)

Step 1) Use the given angle to find the height of the triangle: h=bsin⁡(A)h=b \sin (A)

Step 2) Check if,
Side  aSide\;a < hh,
then no triangles
Side  a=hSide\;a=h,
then 1 triangle
Side  aSide\;a > hh,
then 1 triangle
hh < Side  aSide\;a < Side  bSide\;b,
then 2 triangles

Step 3) Solve the triangle(s)!
Concept

Introduction to the Law of Sines

The Law of Sines is a fundamental concept in trigonometry that plays a crucial role in solving triangles. This powerful mathematical tool allows us to find unknown sides or angles in a triangle when we have limited information. Our introduction video provides a clear and concise explanation of this important trigonometric principle, making it easier for students to grasp its significance. By understanding the Law of Sines, learners can tackle a wide range of problems in geometry, physics, and engineering. The video demonstrates how to apply this law in various scenarios, from simple right-angled triangles to more complex oblique triangles. Mastering the Law of Sines is essential for anyone studying trigonometry or pursuing fields that involve spatial reasoning and calculations. With this knowledge, students can confidently approach triangle-related problems and develop a deeper understanding of trigonometric relationships.

The Law of Sines is particularly useful when dealing with non-right triangles. It provides a way to find unknown sides or angles using the ratios of the sides and the sines of their opposite angles. This method is not only applicable in theoretical mathematics but also in practical fields such as physics and engineering, where understanding the properties of triangles is essential. By mastering the Law of Sines, students can enhance their problem-solving skills and gain a deeper appreciation for the beauty and utility of trigonometric relationships.

Example

Given the following triangle △ABC\triangle ABC,
Using law of sines to find angles and side lengths of triangles
Solve for ∠C\angle C

Step 1: Understanding the Law of Sines

In this question, we are going to solve for the angle CC. To solve for angle CC, we need to use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it is expressed as: asin⁡A=bsin⁡B=csin⁡C \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} where aa, bb, and cc are the lengths of the sides opposite to angles AA, BB, and CC respectively.

Step 2: Identifying the Given Information

To use the Law of Sines, we need at least one pair of a side length and its opposite angle. In the given triangle △ABC\triangle ABC, we need to check if we have this information. We are given:

  • Angle B=62∘B = 62^\circ
  • Side b=8.4b = 8.4 (opposite to angle BB)
  • Side c=6.3c = 6.3 (opposite to angle CC)
We have a pair: angle BB and side bb. Additionally, we have the length of side cc.

Step 3: Setting Up the Law of Sines

Since we are solving for angle CC, it is more efficient to use the version of the Law of Sines with the angles on top: sin⁡Bb=sin⁡Cc \frac{\sin B}{b} = \frac{\sin C}{c} Substituting the known values, we get: sin⁡62∘8.4=sin⁡C6.3 \frac{\sin 62^\circ}{8.4} = \frac{\sin C}{6.3}

Step 4: Solving for sin⁡C\sin C

To isolate sin⁡C\sin C, we multiply both sides of the equation by 6.3: sin⁡C=6.3×sin⁡62∘8.4 \sin C = \frac{6.3 \times \sin 62^\circ}{8.4} Using a calculator, we find: sin⁡62∘≈0.8829 \sin 62^\circ \approx 0.8829 Therefore: sin⁡C=6.3×0.88298.4≈0.6615 \sin C = \frac{6.3 \times 0.8829}{8.4} \approx 0.6615

Step 5: Finding Angle CC

To find angle CC, we take the inverse sine (arc sine) of 0.6615: C=sin⁡−1(0.6615) C = \sin^{-1}(0.6615) Using a calculator, we find: C≈41.47∘ C \approx 41.47^\circ Therefore, angle CC is approximately 41.47∘41.47^\circ.

FAQs

Here are some frequently asked questions about the Law of Sines:

1. How do I use the law of sines to solve problems?

To use the Law of Sines, follow these steps:

  1. Identify the known sides and angles in your triangle.
  2. Set up the Law of Sines equation: a/sin(A) = b/sin(B) = c/sin(C).
  3. Plug in the known values and solve for the unknown side or angle.
  4. Use your calculator to compute the final answer.

2. What is an example of the sine law?

Here's a simple example: In triangle ABC, if angle A = 30°, angle B = 45°, and side a = 10 cm, we can find side b using the Law of Sines. The equation would be: 10/sin(30°) = b/sin(45°). Solving this, we get b 12.47 cm.

3. What is the problem with the sine rule?

The main issue with the Law of Sines is the ambiguous case. This occurs when solving an SSA (Side-Side-Angle) triangle, where two sides and a non-included angle are known. In this scenario, there may be zero, one, or two possible solutions, requiring careful analysis to determine the correct answer.

4. What is a real life example of the law of sines?

A real-life application of the Law of Sines is in navigation. For instance, a ship's captain can use it to determine the distance to a lighthouse. By measuring two angles from different positions on the ship and knowing the distance between these positions, the captain can calculate the distance to the lighthouse using the Law of Sines.

5. How does the Law of Sines differ from the Law of Cosines?

While both laws are used to solve triangles, they have different applications. The Law of Sines is typically used when we know two angles and one side (AAS) or two sides and a non-included angle (SSA). The Law of Cosines is used when we know three sides (SSS) or two sides and the included angle (SAS). The Law of Cosines involves squaring side lengths, while the Law of Sines uses only ratios.

Prerequisites

Mastering the Law of Sines requires a solid foundation in several key trigonometric concepts. Understanding these prerequisite topics is crucial for students to grasp the full scope and application of this important trigonometric law.

One of the fundamental skills needed is the ability to use sine ratio to calculate angles and sides in right triangles. This forms the basis for understanding how the Law of Sines extends these principles to non-right triangles. Similarly, familiarity with the cosine ratio is essential, as it complements the sine ratio in trigonometric calculations.

Students should also be comfortable with various combinations of SohCahToa questions, which reinforce the relationships between sides and angles in triangles. This knowledge is directly applicable when using the Law of Sines to solve more complex triangular problems.

A strong grasp of trigonometric identities is also beneficial. While not directly used in the Law of Sines, understanding these identities enhances overall trigonometric proficiency and problem-solving skills.

For practical applications, students should be familiar with word problems relating to angles in trigonometry. This helps in recognizing real-world scenarios where the Law of Sines can be applied. Additionally, understanding angles of elevation and depression is crucial for many practical applications of the Law of Sines, especially in surveying and navigation problems.

While more advanced, knowledge of inverse trigonometric functions can provide a deeper understanding of the relationships between angles and sides in triangles, which is at the core of the Law of Sines.

By mastering these prerequisite topics, students will be well-prepared to tackle the Law of Sines. This law is a powerful tool in trigonometry, allowing for the solution of non-right triangles and opening up a wide range of applications in physics, engineering, and navigation. The strong foundation provided by these prerequisites ensures that students can not only understand the Law of Sines but also appreciate its significance and versatility in solving complex trigonometric problems.