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Angle Formula

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Understanding Angle Formulas and Trigonometry

Comprehensive Definition, Description, Examples & Rules 

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A branch of mathematics that helps us find the relationship between the angles and sides of triangles is called trigonometry. As an important mathematical operation, trigonometry plays an integral part in miscellaneous domains like engineering, physics, navigation, etc To get to know trigonometry fully, one needs to understand varied elemental ideas like sides, angles, and unit circles, along with important trigonometric functions like sine, cosine, tangent, etc:

Angles

An angle is just a dimension of the amount of rotation between two rays that share a common endpoint, called the vertex. Angles are mostly measured in degrees (°) or radians (rad). Radians are more used in angle formulas in trigonometry because they are more convenient when performing mathematical operations. A full circle is analogous to 360 degrees or 2π radians.

Sides of a Right Triangle

Trigonometry uses right triangles with one angle equal to 90 degrees. In a right-angle triangle, there are three basic sides:

  •         Hypotenuse: The side that lies opposite to the right angle is known as the hypotenuse and is the longest side in a right-angle triangle.
  •         Adjacent Side: The side of the triangle that is adjacent to the angle being considered is known as the adjacent side.
  •         Opposite Side: The side of the triangle opposite to the considered angle is known as the adjacent side.

Unit Circle

A circle with a radius of 1 unit centred at the origin of a coordinate system is known as the unit circle. Every point on the circle refers to an angle, and the coordinates of that point represent values of the sine and cosine of the angle.

For an angle θ in a position where the first side of the angle starts along the positive x-axis, the coordinates of the point lying on the unit circle corresponding to θ are (cos (θ), sin (θ)).

Trigonometric Functions

Trigonometric functions are mathematical operations that connect a right triangle’s angles to its sides’ proportions. The three standard trigonometric functions are sine, cosine and tangent:

  •         Sine (SIN): The sine of an angle in a right triangle is the ratio of the length of the side opposite of that angle to the length of the hypotenuse.

SIN θ = opposite/ hypotenuse

  •         Cosine (COS): The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to that angle to the length of the hypotenuse.

COS θ = adjacent/ hypotenuse

  •         Tangent (TAN): The tangent of a right triangle is the ratio of the length of the side opposite of that angle to the length of the side adjacent to that angle.

TAN θ = opposite/adjacent

The Cosine Rule and Calculating Angles

The cosine rule to find angles is an important trigonometric formula used to find measures of angles in triangles when you know the lengths of two sides and the included angle. It comes in handy when trying to calculate angles in non-right triangles.

Formula for Cosine Rule

In a triangle with sides of lengths a, ‘b’, ‘c’, and an angle ‘C’ opposite side ‘c’, the cosine rule stands as follows:

 c^2 = a^2 + b^2 – 2ab*cos(C)

Here’s how you can use the cosine rule to find an angle when you have two sides and the included angle:

  • Identify the lengths of two sides of the triangle (a and b) and the length of the included angle (C).
  • Employ these numerals in the formula above. 
  • You can rearrange the formula equation to find the included angle (C).

Calculating Angles from Two Sides

To know the application of the cosine rule and understand how to find angles when only two sides of a triangle are known, let’s go through the following example:

Find the measure of angle A in a triangle with side a= 8 units, b= 10 units and the measure of the included angle C as 45 degrees.

Solution:

1)    Put the given values in the cosine rule formula;

c^2 = a^2 + b^2 – 2ab*cos(C)

Here,

A= 8

B= 10

C= 45

2)    Rearrange to solve for A

A= COS^-1 (a^2+b^2-c^2/2ab)

Hence, COS A= a^2+b^2-c^2/2ab

3)    Substitute the given values

So, COS A= 8^2+10^2-c^2/2*8*10

COS A= 64+100-C^2/160

COS A= 164-C^2/160

COS A= 0.707

The degree corresponding to the value COS A= 0.707 is 45 degrees. So, the measure of angle A is approximately 45 degrees.

Finding Angles of a Triangle Using Trigonometry

Trigonometry is a very useful mathematical operation that helps us find the measures of angles when we have the information on the lengths of the sides by applying trigonometric functions like sine, cosine, and tangent along with inverse trigonometric functions. To understand how to find the angles of a triangle, two alternative methods used for calculating angles in such cases are the sine rule and tangent rule:

Sine Rule

The sine rule is used when you know the lengths of two sides of a right triangle and the measure of an angle that does not fall between the given sides. It relates to the ratio of the lengths of the sides to the sines of their opposite angles. The sine rule is mathematically represented as:

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

Where a, b, and c represent the dimension of sides.

A, B, and C are measurements of the angles opposite the individual sides.  

Tangent Rule

If you know the measurements of two sides and the value of the included angle, i.e., the angle that falls between the given sides, the tangent rule can be used as the tangent of half the difference of the angles formed by the given sides corresponds to the tangent of half the sum of those angles. The tangent rule is represented as:

Tan (A-B/2)/Tan (A+B/2)= a-b/a+b= c-d/c+d

Where A and B are used to denote angles formed by the given sides

A and b denote the measurement of the lengths of the sides. 

C and d represent the measurement of the lengths of sides.

Methodical Approach to Finding Angles

If you want to know the process of finding angles of a triangle, here is a systematic approach to solving angle-related problems using trigonometric rules:

1)    Look for side lengths and angle measures and check whether you have all three sides, two sides and an included angle, or two sides and a non-included angle.

2)    Determine what you need to compute.

3) Select the appropriate method based on the given information:

  • Use the cosine rule to find one of the angles if you have the measurement of all three sides. 
  • Use the sine rule to find one of the other angles if you have the measurement of two sides and an included angle.
  • Use the tangent rule to find one of the other angles if you have the measurement of two sides and a non-included angle.

Real-Life Applications

Trigonometry is important in a variety of real-world scenarios, such as:

Architects can develop safer structures for us that are also solid and durable only through the use of angles in trigonometry.

Navigators get to know about their precise location and directions using trigonometric functions and computing angles. 

Research and survey about land is possible using trigonometric formulas to calculate entities like heights, lengths and angles. 

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Key Takeaways

  1. Trigonometry helps us determine the relationship between the angles and sides of triangles.

  2. The three rudimentary trigonometric functions are sine, oscine, and tangent.

  3. Trigonometry is used in a range of real-world scenarios like navigation, architecture and conducting surveys.

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Frequently Asked Questions

You can find the missing angle in a triangle by making use of the known fact that the sum of all angles in a triangle is 180 degrees. 

To find angles in a quadrilateral, you need to minus the known angles from 360 degrees in the case of a convex quadrilateral and 180 degrees in the case of a concave quadrilateral. 

To calculate the angles in a circle or a sector, you should follow the formula: Angle= (Aec length/ circle circumference)* 360.

To calculate angles in a right triangle using trigonometry, you can make use of sine, cosine and tangent functions according to the provided information. 

The formula for calculating the exterior angles of a polygon is Exterior Angle = 360 degrees / Number of sides.

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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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