# Area of a triangle using sine and cosine rule

## Proofs of sine rule, cosine rule, area of a triangle 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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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:.

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. Therefore we use the Sine Rule.

The area rule states that the area of any triangle is equal to half the product of the . Use your results to write a general formula for the sine rule given ?PQR.

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,.

In trigonometry , the law of cosines also known as the cosine formula , cosine rule , or Al-Kashi's theorem  relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. For the same figure, the other two relations are analogous:. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. Though the notion of the cosine was not yet developed in his time, Euclid 's Elements , dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines.

## Section 4: Sine And Cosine Rule

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., 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.

## Cosine Rule and Area of Triangle

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. Sine rule, cosine rule and area of triangle

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

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## 1 thoughts on “Area of a triangle using sine and cosine rule”

1. Luce L. says:

This rule also holds if we use angles B or C and sides b or c instead of angle A and side a.