Geometry and Trigonometry Formula Review

Geometry: Polygons and Their Properties

  • Diagonal Calculation in Polygons:

    • The number of diagonals in a polygon with nn sides is given by the formula:     n(n−3)2\frac{n(n-3)}{2}

  • Interior and Exterior Angles:

    • Sum of Interior Angles: The total sum of all interior angles for any polygon with nn sides is calculated as:     (n−2)×180∘(n - 2) \times 180^{\circ}
    • Measure of Each Interior Angle: For a regular polygon, the measure of each individual interior angle is:     (n−2)×180∘n\frac{(n - 2) \times 180^{\circ}}{n}
    • Sum of Exterior Angles: The sum of all exterior angles for any convex polygon is always constant:     360∘360^{\circ}
    • Measure of Each Exterior Angle: For a regular polygon, the measure of each individual exterior angle is:     360∘n\frac{360^{\circ}}{n}

Business Mathematics: Profit and Loss

  • Overall Profit or Loss for Two Articles Sold at the Same Selling Price (SP):
    • Scenario: Two articles are sold at the exact same selling price.
    • Conditions: The first article is sold at a profit of x%x\%, and the second article is sold at a profit or loss of y%y\%.
    • Formula: The overall percentage of profit or loss can be determined using the following expression:     100(x+y)+2xy200+x+y\frac{100(x + y) + 2xy}{200 + x + y}

Algebra: Polynomial Identities

  • Relationship Involving Cubic Sums:
    • If the quadratic equation is given as x2+1=3xx^2 + 1 = \sqrt{3}x, it can be rearranged by dividing all terms by xx to find:     x+1x=3x + \frac{1}{x} = \sqrt{3}
    • Cubing both sides of the equation x+1x=3x + \frac{1}{x} = \sqrt{3} results in:     (x+1x)3=(3)3(x + \frac{1}{x})^3 = (\sqrt{3})^3x3+1x3+3(x+1x)=33x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x}) = 3\sqrt{3}
    • Substituting the value of (x+1x)(x + \frac{1}{x}) back into the expression:     x3+1x3+33=33x^3 + \frac{1}{x^3} + 3\sqrt{3} = 3\sqrt{3}
    • This leads to the identities:     x3+1x3=0x^3 + \frac{1}{x^3} = 0x6+1=0x^6 + 1 = 0x6=−1x^6 = -1

Trigonometry: Relative Values of Sine and Cosine

  • Comparison Within the First Quadrant (0∘<θ<90∘0^{\circ} < \theta < 90^{\circ}):
    • Case 1: When the angle θ\theta is between 45∘45^{\circ} and 90∘90^{\circ} (45∘<θ<90∘45^{\circ} < \theta < 90^{\circ}):     sin⁡(θ)>cos⁡(θ)\sin(\theta) > \cos(\theta)
    • Case 2: When the angle θ\theta is between 0∘0^{\circ} and 45∘45^{\circ} (0∘<θ<45∘0^{\circ} < \theta < 45^{\circ}):     cos⁡(θ)>sin⁡(θ)\cos(\theta) > \sin(\theta)

Trigonometry: Key Identities and Transformations

  • Square Root Identities for Sum and Difference:

    • 1+sin⁡(2θ)=sin⁡(θ)+cos⁡(θ)\sqrt{1 + \sin(2\theta)} = \sin(\theta) + \cos(\theta)
    • 1−sin⁡(2θ)=sin⁡(θ)−cos⁡(θ)\sqrt{1 - \sin(2\theta)} = \sin(\theta) - \cos(\theta) (if sin⁡(θ)>cos⁡(θ)\sin(\theta) > \cos(\theta)) or cos⁡(θ)−sin⁡(θ)\cos(\theta) - \sin(\theta) (if cos⁡(θ)>sin⁡(θ)\cos(\theta) > \sin(\theta)).

  • Ratio Transformations involving Tangent:

    • The ratio of the sum and difference of sine and cosine can be expressed in terms of tangent:     sin⁡(θ)+cos⁡(θ)sin⁡(θ)−cos⁡(θ)=tan⁡(θ)+1tan⁡(θ)−1=1+tan⁡(θ)tan⁡(θ)−1\frac{\sin(\theta) + \cos(\theta)}{\sin(\theta) - \cos(\theta)} = \frac{\tan(\theta) + 1}{\tan(\theta) - 1} = \frac{1 + \tan(\theta)}{\tan(\theta) - 1}
    • cos⁡(θ)+sin⁡(θ)cos⁡(θ)−sin⁡(θ)=1+tan⁡(θ)1−tan⁡(θ)=tan⁡(45∘+θ)\frac{\cos(\theta) + \sin(\theta)}{\cos(\theta) - \sin(\theta)} = \frac{1 + \tan(\theta)}{1 - \tan(\theta)} = \tan(45^{\circ} + \theta)
    • cos⁡(θ)−sin⁡(θ)cos⁡(θ)+sin⁡(θ)=1−tan⁡(θ)1+tan⁡(θ)=tan⁡(45∘−θ)\frac{\cos(\theta) - \sin(\theta)}{\cos(\theta) + \sin(\theta)} = \frac{1 - \tan(\theta)}{1 + \tan(\theta)} = \tan(45^{\circ} - \theta)
    • sin⁡(θ)−cos⁡(θ)sin⁡(θ)+cos⁡(θ)=tan⁡(θ)−1tan⁡(θ)+1\frac{\sin(\theta) - \cos(\theta)}{\sin(\theta) + \cos(\theta)} = \frac{\tan(\theta) - 1}{\tan(\theta) + 1}

  • Product-Difference Equivalence:

    • tan⁡2(θ)−sin⁡2(θ)=tan⁡2(θ)×sin⁡2(θ)\tan^2(\theta) - \sin^2(\theta) = \tan^2(\theta) \times \sin^2(\theta)
    • cot⁡2(θ)−cos⁡2(θ)=cot⁡2(θ)×cos⁡2(θ)\cot^2(\theta) - \cos^2(\theta) = \cot^2(\theta) \times \cos^2(\theta)

  • Double Angle Representations in terms of Tangent:

    • sin⁡(2θ)=2tan⁡(θ)1+tan⁡2(θ)\sin(2\theta) = \frac{2\tan(\theta)}{1 + \tan^2(\theta)}
    • cos⁡(2θ)=1−tan⁡2(θ)1+tan⁡2(θ)\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}

Trigonometry: Compound Angle Formulas

  • Tangent Compound Angles:

    • The tangent of the sum of two angles is:     tan⁡(A+B)=tan⁡(A)+tan⁡(B)1−tan⁡(A)tan⁡(B)\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}
    • The tangent of the difference of two angles is:     tan⁡(A−B)=tan⁡(A)−tan⁡(B)1+tan⁡(A)tan⁡(B)\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}

  • Special Case for Tangent Angles:

    • If A+B=45∘A + B = 45^{\circ}, then:     (1+tan⁡(A))(1+tan⁡(B))=2(1 + \tan(A))(1 + \tan(B)) = 2
    • If A+B=135∘A + B = 135^{\circ}, then:     (1−tan⁡(A))(1−tan⁡(B))=2(1 - \tan(A))(1 - \tan(B)) = 2