# Area of a triangle using sine and cosine rule

## Section 4: Sine And Cosine Rule

The Law of Sines (Sine Rule) and Cosine Rule GCSE Maths revision section of Revision The area of any triangle is ? absinC (using the above notation).

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This rule also holds if we use angles B or C and sides b or c instead of angle A and side a. This leads to the following rule called the Sine Rule:. This is the basic rule used in land measurement and map making. We can rewrite the rule in three parts to use depending on the sides and angles that are known. The following diagrams show the same triangle standing on each of the three sides in turn.

In the module Further trigonometry Year 10 , we introduced and proved the sine rule , which is used to find sides and angles in non-right-angled triangles. In general, one of the three angles may be obtuse. The formula still holds true, although the geometric proof is slightly different. In the module Congruence Year 8 , it was emphasised that, when applying the SAS congruence test, the angle in question has to be the angle included between the two sides. Detailed description of diagram. There are two non-congruent triangles that satisfy the given data. This situation is sometimes referred to as the ambiguous case.

You will need to know at least one pair of a side with its opposite angle to use the Sine Rule. Practice Questions Work out the answer to each question then click on the button marked to see if you are correct. Finding Sides If you need to find the length of a side, you need to know the other two sides and the opposite angle. Sides b and c are the other two sides, and angle A is the angle opposite side a. These examples illustrate the decision-making process for a variety of triangles: e. We do know a side and its opposite angle.

Pythagoras' Theorem describes the mathematical relationship between three sides of a right-angled triangle. Pythagoras' Theorem states that; in a right-angled triangle the square of the hypotenuse longest side is equal to the sum of the squares of the other two sides. It is written in the formula:. As well as Pythagoras' Theorem, there are other formulae which can be used to calculate a unknown side or angle in a triangle; such as trigonometry. These functions are defined as the ratios of the different sides of a triangle. The functions of sin, cos and tan can be calculated as follows:.

Substituting the value of h in the formula for the area of a triangle, you get. Similarly, you can write formulas for the area in terms of sin B or sin C. You have the lengths of two sides and the measure of the included angle. Use the Pythagorean Theorem to find the length of the third side of the triangle. Given that the angle at the vertex Y is a right angle. Using the Triangle Angle Sum Theorem , the measure of the third angle is,. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.

## Trigonometry - Sine and Cosine Rule

Area Of A Non-Right Angle Triangle

## What is the Formula for the Area of a Triangle Using Sines?

The solution for an oblique triangle can be done with the application of the Law of Sine and Law of Cosine, simply called the Sine and Cosine Rules. An oblique triangle, as we all know, is a triangle with no right angle. It is a triangle whose angles are all acute or a triangle with one obtuse angle. The Sine Rule states that the sides of a triangle are proportional to the sines of the opposite angles. In symbols,. In this case, there may be two triangles, one triangle, or no triangle with the given properties. For this reason, it is sometimes called the ambiguous case.

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## Trigonometry and Pythagoras

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1. Witchmemnamo says:

In trigonometry , the law of cosines also known as the cosine formula , cosine rule , or Al-Kashi's theorem [1] relates the lengths of the sides of a triangle to the cosine of one of its angles.

2. Imperio A. says:

3. Agar D. says: