# When to Use the Law of Sines and Cosines

You probably learned the basic trigonometric functions in school – sine, cosine, and tangent. But do you remember when to use each one?

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## Introduction

In trigonometry, there are a few different ways to find missing sides or angles in a triangle. Two of the most common methods are using the law of sines and the law of cosines. But when should you use each method? In this article, we’ll explore when it’s appropriate to use the law of sines and cosines so that you can solve any triangle problem that comes your way.

## What is the law of sines?

In mathematics, the law of sines, sine rule, sine formula, or sine relation is an equality relating the lengths of certain sides of a triangle to the sines of its angles. According to the law,

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

where a, b and c are the lengths of the sides of the triangle and A, B and C are the angles opposite those respective sides. The law can be applied to any triangle, whether it is right angled or not. It can also be extended to non-right angled triangles if one replaces sin(A), sin(B), sin(C) by the so-called directional cosines which are in fact the ratios of certain sides in a corresponding reference triangle.

## What is the law of cosines?

The law of cosines is used to find the side lengths or angles of a triangle when we know some or all of the other side lengths or angles. This can be really useful in many different geometry and trigonometry problems. The law of cosines is also sometimes called the cosine rule.

## How to use the law of sines

The law of sines and cosines are two related formulas that are often used together to solve problems. They are both ways of finding missing sides or angles in a triangle, when you know some of the other sides or angles.

To use the law of sines, you need to know:
– The length of one side of the triangle
– The angle opposite that side
– The other two angles in the triangle

To use the law of cosines, you need to know:
– The lengths of two sides of the triangle
– The angle between those two sides
– Either:
--the length of the third side
--or one of the other two angles in the triangle

## How to use the law of cosines

There are two main trigonometric functions- Sine and Cosine. These functions help us to calculate the missing sides and angles of a triangle when we only know some of the information. The law of sines and cosines are both very useful, but it can be tricky to know when to use each one. In this article, we’ll go over when to use the law of cosines so that you can tackle any trigonometry problem with confidence!

The law of cosines states that:

c^2 = a^2 + b^2 – 2ab * cos(C)

This means that if you know the lengths of two sides of a triangle, as well as the angle between them (angle C in the equation), you can use the law of cosines to find the length of the third side. This is particularly useful when you’re working with a oblique triangle- a triangle where none of the angles are 90 degrees.

To use the law of cosines, simply plug in the known values for a, b, and C into the equation. Then, solve for c. This will give you the missing side length that you were looking for!

## Applications of the law of sines

The law of sines is typically used in two scenarios: when you know two angles and the length of one side of a triangle (AAS), or when you know two sides and the angle between them (SAS). We will explore these scenarios in greater depth below.

AAS: When you know two angles and the length of one side of a triangle, you can use the law of sines to find the remaining lengths. This is helpful when you are trying to find the height of an object, for example.

To use the law of sines in this scenario, you will set up your equation as follows:

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

In this equation, a is the known side, B and C are the known angles, and b and c are the unknown sides. You can solve for either unknown side using this equation.

SAS: When you know two sides and the angle between them, you can use the law of cosines to find all three missing parts. This is helpful when you are trying to find an object that is hidden from view.

To use the law of cosines in this scenario, you will set up your equation as follows:
a^2 = b^2 + c^2 – 2bc*cos(A)

## Applications of the law of cosines

There are many applications for the law of cosines, including navigation, astronomy, and solving triangles. In navigation, the law of cosines can be used to find the distance between two points on a sphere, such as two ships at sea. In astronomy, the law of cosines can be used to find the distance to a star or other celestial objects. Finally, the law of cosines can be used to solve any triangle, whether it is right angled or not.

## Advantages of using the law of sines

There are many different situations in which you might need to use the law of sines or cosines. For example, you might need to find the height of an object when you only know the angles of the triangle formed by the object and the ground, or you might need to find the length of a side of a triangle when you only know the lengths of two other sides. In general, you can use either the law of sines or cosines whenever you known two angles and one side (or two sides and one angle) of a triangle, but it’s sometimes more convenient to use one over the other.

One advantage of using the law of sines is that it doesn’t require any information about angles that are not adjacent to the side you’re trying to find. So, if you know angle A and side a, and you want to find angle B, you can just use the law of sines without having to know angle C. This can be helpful if angle C is difficult to measure or if you don’t have all of the information about the triangle.

Another advantage of using the law of sines is that it sometimes yields simpler calculations than using the law of cosines. This can be particularly helpful if you’re working with large numbers or decimals.

However, there are also some advantages to using the law of cosines. One is that it can be used in cases where one angle is obtuse (greater than 90 degrees). The law of sines can only be used with acute angles (angles less than 90 degrees). Another advantage is that it sometimes yields more accurate results than using the law
of sines, especially when working with small numbers or decimals.

Ultimately, which method you use is up to you – both have their advantages and disadvantages.

## Advantages of using the law of cosines

There are many mathematical formulas that can be used to solve certain types of problems. Some formulas are more useful than others in specific situations. The law of cosines is a good example of a formula that has specific advantages and disadvantages.

The law of cosines is a mathematical formula that can be used to solve certain problems involving triangles. The formula is derived from the Pythagorean theorem, which states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.

The law of cosines can be used to find the length of a side of a triangle when two other sides and the angle between them are known. This is sometimes more accurate than using the Pythagorean theorem alone. In addition, the law of cosines can be used to find angles when two sides and one angle are known. This can be very useful in certain situations, such as when working with surveying data.

One disadvantage of using the law of cosines is that it can be more difficult to remember than some other formulas. In addition, the law of cosines only works with triangles, so it cannot be used to solve problems involving other shapes.

## Conclusion

We hope you enjoyed this lesson on the law of sines and cosines. Now that you know when and how to use these formulas, you’ll be able to solve many types of problems involving triangles. Be sure to practice what you’ve learned so that you can master these concepts.

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