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Cosine Formula

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Exploring the Cosine Rule: Formulas, Applications, and Angle Calculations

Comprehensive Definition, Description, Examples & Rules 

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Introduction

The cosine­ rule is a mathematical formula that connects the­ lengths of the sides of a triangle­ to one of the angles in that triangle­, specifically the cosine angle­. This formula allows for calculating the length of a specific side­ or determining the me­asure of an angle when give­n its corresponding side. Additionally, it’s possible to use­ this rule effective­ly by using the hypotenuse to calculate­ the cosine angle.

Significance in Geometry

The importance of the cosine rule in geometry is:

  • It helps to calculate the length of a side of a particular triangle in geometry, and you can also find the measure of each side of the triangle.
  • It is also effective enough to calculate and indicate the bigger side of the triangle and use the hypotenuse rule effectively. 

Significance in Trigonometry

The importance of the cosine rule in trigonometry is:

  • The rule allows you to find the angles of the triangle and determine each side. 
  • It helps to make trigonometrical calculations and find the length of the side respective of all the given triangles.

Understanding the Cosine Rule

The fundamental concept of the cosine rule is that it is used to determine the length of a particular side of a triangle and also can determine the cosine angle of the triangle using the hypotenuse rule. 

The cosine rule definition is you need two sides of the triangle and one of the opposite side, and you can use them to find out essential details of the triangle. As mentioned earlier, the importance of the cosine rule in trigonometry determines the particular angle and helps make the other trigonometrical calculations possible. 

You can use the cosine rule when you want to determine the particular side of a triangle, or in situations when you want to determine a specific angle of the triangle, then you can use the cosine rule effectively. 

The Cosine Rule Formula

The formula to calculate the cosine rule comes with a General form. Then, there is a similar formula variation, which helps calculate the triangle’s sides and angle.

The general formula is:

  • a2 = b2 + c2 – 2bc·cosA

The variation in the formula is:

  • b2 = c2 + a2 – 2ca·cosB
  • c2 = a2 + b2 – 2ab·cosC

Here, A, B, and C are the vertices of the triangle, and their opposite sides are represented in the form of a, b, and c. 

Example

Two sides of a triangle measure 72 in and 50 in, with the angle between them measuring 49º let us find the missing side.

Solution:

Substitute 72 for b, 50 for c and 49º for A.

Using the law of cosines formula,

a2 = b2 + c2 – 2bc·cosA

a2 = (72)2 + (50)2 – 2(72)(50)cos49º

a2 = 5184 + 2500 – (7200)(0.656)

a2 = 5184 + 2500 – 4723.2

a2 = 2960.8

a ≈ 54.4

So, the missing length of the side is 54.4.

Applications of the Cosine Rule

The practical applications of the cosine rule in real-life scenarios happen in:

  • You can use this law to calculate the distance of travelling.

In special fields:

  • Navigation: It is also effective in making trigonometric cosine rule calculations for pilots and sailors as it helps in navigation 
  • Engineering: While calculating the distance or the angles of the elevation in situations when constructing telephone poles and bridges.
  •  Physics: The physical equation used the trigonometric cosine rule; many laws of Newton have used theory during calculation. 

The case studies to check are:

  • KS Leung: Mathematical Gazette
  • N Bibby: Cosine Rule Theorem

Using the Cosine Rule to Calculate Angles

The special Cosine rule to calculate the angle is that you need to find the length of a given side, and then you should apply the cosine rules formula and determine the missing angle. 

Example:

Find the angle opposite to the side BC of the Triangle where sides are AB 5 inches, AC 7 inches, and BC 10 inches.

We need to find the measure of ∠A.

We will use the formula

a2 = b2 + c2 – 2bc.cosA. Substitute 10 for ‘a’, 7 for ‘b’ and 5 for ‘c’.

102 = (7)2 + (5)2 – 2(5)(7)·cosA

70·cos A = -26

cos A = -13/35

A = 111.8º

The special consideration while using the rule for angles is to determine the sides first and then use the calculations appropriately. 

Tips and Tricks

The tips and tricks which will help to apply the cosine rules effectively are:

  • Determine all the sides first, which will help to calculate the cosine rule effectively and use the formula properly, which is required. 
  • The calculation should be in terms of the sign mentioned in the question, and whether it is the missing angle or the missing side is up to you to determine. 

Common Mistakes and Challenges

The various mistakes and challenges that you should avoid while using the cosine rule are:

  • When there are only two sides and the included angle, but you need help finding the third side, the formula can be challenging to calculate. 
  • When the answer to your question comes from an unknown angle, it will also be challenging to find your answer. 

There are three forms of the cosine rule, and you should effectively determine the one appropriate for your calculation. Check the details of the formula, and you will be easily able to determine the appropriate formula. 

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Key Takeaways

  1. The Cosine Rule is effective enough to determine and calculate the length of a particular side of a triangle or a specific angle of a triangle. 
  2. The Cosine Rule has a general formula and a different form, and you can use the forms to calculate the particular length of the side or the angle of the triangle. 
  3. The Pythagoras Theorem is also effective when using the formula to calculate the angle of a triangle. 

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Frequently Asked Questions

The different forms of the Cosine Rule are:

The general formula is:

  • a2 = b2 + c2 – 2bc·cosA

The variation in the formula is:

  • b2 = c2 + a2 – 2ca·cosB
  • c2 = a2 + b2 – 2ab·cosC

Yes, you can easily apply the cosine rule for all angle triangles when using any particular triangle length is not compulsory. 

The cosine rule for angle is:

  • Check all the provided data and determine all three sides of the triangle as if you are to measure an angle, then two of the sides will be provided from which you can calculate the third side using the Pythagoras theorem. 
  • Then, apply the cosine formula to express your obtained data. 

The only particular case of consideration while using the cosine rule is when using the cosine rule to determine the angle, and you have to use the Pythagoras theorem to determine it. 

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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