Properties-and-solution-of-triangle Formula Sheet

Inverse trigonometric functions reverse the usual ratios, allowing you to find angles when given a ratio. They include functions like arcsin, arccos, and arctan, which map values from the ratio back to an angle in specific ranges. Think of them as the GPS that gets you back to the angle when you’ve wandered too far into sine, cosine, or tangent territory!

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1. Sine Formula:

    In any triangle, ABC

$\space \space \space \space \space \space \frac{a}{sin \space A} = \frac{b}{sin \space B} = \frac{c}{sin \space C} = \lambda = \frac{abc}{2\Delta} = 2R$

$\space \space \space \space \space$ where R is circumradius and $\Delta$ is area of triangle.

2. Cosine Formula:

$\space \space \space \space \space \space$ (a) $cos \space A = \frac{b^2+c^2-a^2}{2bc} \space or \space a^2 = b^2 + c^2 - 2bc \space cos A$

$\space \space \space \space \space \space$ (b) $cos \space B = \frac{c^2+a^2-b^2}{2ca}$

$\space \space \space \space \space \space$ (c) $cos \space C = \frac{a^2+b^2-c^2}{2ab}$

3. Projection Formula:

$\space \space \space \space \space \space$ (a) $b \space cos \space C + c \space cos \space B = a$

$\space \space \space \space \space \space$ (b) $c \space cos \space A + a \space cos \space C = b$

$\space \space \space \space \space \space$ (c) $a \space cos \space B + b \space cos \space A = c$

4. Napier's Analogy (Tangent Rule):

$\space \space \space \space \space \space$ (a) $tan \big\lgroup \frac{B-C}{2} \big\rgroup = \frac{b-c}{b+c} cot \frac{A}{2}$

$\space \space \space \space \space \space$ (b) $tan \big\lgroup \frac{C-A}{2} \big\rgroup = \frac{c-a}{c+a} cot \frac{B}{2}$

$\space \space \space \space \space \space$ (c) $tan \big\lgroup \frac{A-B}{2} \big\rgroup = \frac{a-b}{a+b} cot \frac{C}{2}$

5. Half Angle Formula:

$\space \space \space \space \space \space \space \space \space \space \space \space s = \frac{a+b+c}{2}$ = semi-perimeter of triangle.

$\space \space \space \space \space \space$ (a) (i) $sin \frac{A}{2} = \sqrt \frac{(s-b)(s-c)}{bc} \space \space \space \space \space$ (ii) $sin \frac{B}{2} = \sqrt \frac{(s-c)(s-a)}{ca}$

$\space \space \space \space \space \space \space \space \space \space \space$ (iii) $sin \frac{C}{2} = \sqrt \frac{(s-a)(s-b)}{ab}$

$\space \space \space \space \space \space \space$ (b) (i) $cos \frac{A}{2} = \sqrt \frac{s(s-a)}{bc} \space \space \space \space \space \space \space \space \space \space$ (ii) $cos \frac{B}{2} = \sqrt \frac{s(s-b)}{ca}$

$\space \space \space \space \space \space \space \space \space \space \space \space$ (iii) $cos \frac{C}{2} = \sqrt \frac{s(s-c)}{ab}$

$\space \space \space \space \space \space \space$ (c) (i) $tan \frac{A}{2} = \sqrt \frac{(s-b)(s-c)}{s(s-a)} = \frac{\Delta}{s(s-a)}$

$\space \space \space \space \space \space \space \space \space \space \space \space$ (ii) $tan \frac{B}{2} = \sqrt \frac{(s-c)(s-a)}{s(s-b)} = \frac{\Delta}{s(s-b)}$

$\space \space \space \space \space \space \space \space \space \space \space \space$ (iii) $tan \frac{c}{2} = \sqrt \frac{(s-a)(s-b)}{s(s-c)} = \frac{\Delta}{s(s-c)}$

$\space \space \space \space \space \space \space$(d) Area of Triangle

$\space \space \space \space \space \space \space \space \space \space \space \space \Delta = \sqrt{s(s-a)(s-b)(s-c)}$

$\space \space \space \space \space \space \space \space =\frac{1}{2}bc \space sin \space A = \frac{1}{2}ca \space sin \space B = \frac{1}{2}ab \space sin \space C$

$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space = \frac{1}{4}\sqrt{2({a^2}{b^2}+{b^2}{c^2}+{c^2}{a^2})-{a^4}-{b^4}-{c^4}}$

6. Radius of the Circumcircle 'R':

The circumcentre is the point of intersection of perpendicular bisectors of the sides, and the distance between the circumcentre & vertex of the triangle is called circumradius 'R'.

$\space \space \space \space \space \space R = \frac{a}{2 \space sin \space A} = \frac{b}{2 \space sin \space B} = \frac{c}{2 \space sin \space C} = \frac{abc}{4\Delta}.$

7. The radius of the Incircle 'r':

The point of intersection of internal angle bisectors is the incentre, and the perpendicular distance of the incentre from any side is called inradius 'r'.

$\space \space \space \space \space \space \space \space r = \frac{\Delta}{s} = (s-a) tan \frac{A}{2} = (s-b) tan \frac{B}{2}$

$\space \space \space \space \space \space \space =(s-c) tan \frac{C}{2} = 4R \space sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}.$

8. Radii of the Ex-Circles:

The point of intersection of two external angles and one internal angle bisector is the excentre, and the perpendicular distance of the excentre from any side is called an exradius. If ${r_1}$ is the radius of the escribed circle opposite to angle A of ${\Delta}ABC$ and so on, then :

$\space \space \space \space \space$ (a) $r_1 = \frac {\Delta}{s-a} = s \space tan \frac{A}{2} = 4R \ sin \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}$

$\space \space \space \space \space$ (b) $r_2 = \frac {\Delta}{s-b} = s \space tan \frac{B}{2} = 4R \ cos \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}$

$\space \space \space \space \space$ (c) $r_3 = \frac {\Delta}{s-c} = s \space tan \frac{C}{2} = 4R \space cos \frac{A}{2} cos \frac{B}{2} sin \frac{C}{2}$

9. Length of Angle Bisector, Median & Altitude:

if ${m_a},{\beta_a} \space \& \space {h_a}$ are the lengths of a median, an angle bisector & altitude from the angle A then,

$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \frac{1}{2} \sqrt {b^2+c^2+2bc \ cos A} = {m_a} = \frac{1}{2} \sqrt {2b^2+2c^2-2a^2}$

$\space \space \space \space \space \space$ and $\space \space$ $\beta{_a} = \frac{2bc \space cos \frac{A}{2}}{b+c}, \space \space \space \space \space \space \space \space \space \space$ ${h_a} = \frac{a}{cot B + cot C}$

$\space \space \space \space \space \space$ Note that $m_a^2 + m_b^2 + m_c^2 = \frac{3}{4} (a^2+b^2+c^2)$

10. Orthocentre and Orthic Triangle:

(a) The point of intersection of altitudes is orthocentre & the triangle KLM, which is formed by joining the feet of the altitudes, is called the orthic triangle.

(b) The distances of the orthocentre from the angular points of the ${\Delta}{ABC}$ are $2R \space cos A, 2R \space cos B$ & $2R \space cos C.$

(c) The distance of orthocentre from sides are $2R \space cos B \space cos C, \space 2R \space cos C \space cos A$ and $2R \space cos A \space cos B$

(d) The sides of the orthic triangle are $a \space cos A ( = R \space sin 2A), b \space cos B (= R \space sin 2B)$ and $c \space cos C (=R \space sin 2C)$ and its angles are $\pi - 2A, \pi - 2B$ and $\pi - 2C$

(e) Circumradii of the triangles $PBC, PCA, PAB$ and $ABC$ are equal.

(f) Area of orthic triangle = $2\Delta \space cosA \space cosB \space cosC = \frac{1}{2}R^2 \space sin2A \space sin2B \space sin2C$

(g) Circumradii of orthic triangle $= \frac{R}{2}$

11. Ex-Central Triangle:

(a) The triangle formed by joining the three excentres ${\Iota_1}, {\Iota_2}$ and ${\Iota_3}$ Of $\Delta ABC$ is called the excentral or excentric triangle.

(b) Incentre ${\Iota}$ of $\Delta ABC$ is the orthocentre of the excentral $\Delta {\Iota_1} {\Iota_2}{\Iota_3}.$

(C) $\Delta ABC$ is the orthic triangle of the $\Delta {\Iota_1} {\Iota_2}{\Iota_3}.$

(d) The sides of the excentral triangle are $4R \space cos \frac{A}{2}, 4R \space cos \frac{B}{2}$ and $4R \space cos \frac{C}{2}$ and its angles are $\frac{\pi}{2} - \frac{A}{2}, \frac{\pi}{2} - \frac{B}{2}$ and $\frac{\pi}{2} - \frac{C}{2}.$

(e) ${\Iota \Iota_1} = 4R \space sin \frac{A}{2}; {\Iota \Iota_2} = 4R \space sin \frac{B}{2}; {\Iota \Iota_3} = 4R \space sin \frac{C}{2}.$

12. The Distance between the Special Points:

(a) The distance between circumcentre and orthocentre $\space \space = R \sqrt{1 -8 cos A cos B cos C}$

(b) The distance between circumcentre and incentre $= \sqrt{R^2 - 2Rr}$

(c) The distance between incentre and orthocentre $\space \space = \sqrt{2R^2 - 4R^2 \space cos A cos B cos C}$

(d) The distance between circumcentre & excentre is
$\Omicron \Iota _1 = R \sqrt{1+8 sin \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}} = \sqrt {R^2 + 2R}{_1}$ & so on.

13. m-n Theorem:

$\space \space \space \space \space \space \space \space (m + n) cot \space \theta = m \space cot \space a - n \space cot \space \beta$
$\space \space \space \space \space \space \space \space(m + n) cot \space \theta = n \space cot \space B - m \space cot \space C.$

14. Important Points:

$\space \space \space \space \space \space \space$ (a) (i) If $a \space cos \space B = b \space cos \space A,$ then the triangle is isosceles.
$\space \space \space \space \space \space \space \space \space \space \space \space$ (ii) If $a \space cos \space A = b \space cos \space B,$ then the triangle is isosceles or right angled.

$\space \space \space \space \space \space \space$ (b) In Right Angle Triangle :

$\space \space \space \space \space \space \space \space \space \space \space \space \space$ (i) $a^2 + b^2 + c^2 = 8R^2$

$\space \space \space \space \space \space \space \space \space \space \space \space \space$ (ii) $cos^2 A + cos^2 B + cos^2 C = 1$

$\space \space \space \space \space \space \space$ (c) In equilateral triangle :

$\space \space \space \space \space \space \space \space \space \space \space \space$ (i) $R = 2r$ $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ (ii) $r_1 = r_2 = r_3 = \frac{3R}{2}$

$\space \space \space \space \space \space \space \space \space \space \space \space$ (iii) $r : R : r_1 = 1 : 2 : 3$ $\space \space \space \space \space$ (iv) $area = \frac{\sqrt3{a^2}}{4}$ $\space \space \space \space \space$ (v) $R = \frac{a}{\sqrt3}$

(d) (i) The circumcentre lies (1) inside an acute angled triangle (2) outside an obtuse angled triangle & (3) at mid point of the hypotenuse of right angled triangle

(ii) The orthocentre of right angled triangle is the vertex at the right angle.

(iii) The orthocentre, centroid & circumcentre are collinear & centroid divides the line segment joining orthocentre & circumcentre internally in the ratio 2 : 1, except in case of equilateral triangle. In equilateral triangle all these centres coincide.

15. Regular Polygon:

Consider a 'n' sided regular polygon of side length 'a'

(a) Radius of incircle of this polygon $r = \frac{a}{2} cot \frac{\pi}{n}$

(b) Radius of circumcircle of this polygon $R = \frac{a}{2} cosec \frac{\pi}{n}$

(c) Perimeter & area of regular polygon Perimeter $= na = 2nr \space tan \frac{\pi}{n} = 2nR \space sin \frac{\pi}{n}$ Area $= \frac{1}{2} n R^2 sin \frac{2\pi}{n} = nr^2 tan \frac{\pi}{n} = \frac{1}{4} na^2 cot \frac{\pi}{n}$

16. Cyclic Quadrilateral:

(a) Quadrilateral $ABCD$ is cyclic if $\angle A + \angle C = \pi = \angle B + \angle C$ (opposite angle are supplementary angles)

(b) Area $= \sqrt{(s-a)(s-b)(s-c)(s-d)},$ where $2s = a+b+c+d$

(c) $cos B = \frac{a^2+b^2-c^2-d^2}{2(ab+cd)}$ & similarly other angles

(d) Ptolemy's theorem : If $ABCD$ is cyclic quadrilateral, then $AC . BD = AB . CD + BC . AD$

17. Solution of Triangle:

Case-I: Three sides are given then to find out three angles use $cos A = \frac{b^2+c^2-a^2}{2bc}, \space cos B = \frac{c^2+a^2-b^2}{2ac}, \space \& \space cos C = \frac{a^2+b^2-c^2}{2ab},$

Case-II : Two sides & included angle are given : Let sides a, b & angle C are given then use $tan \frac{A-B}{2} = \frac{a-b}{a+b} cot \frac{C}{2}$ and find value of $A-B$ ......(i)
$\& \space \frac{A+B}{2} = 90 \degree - \frac{C}{2}$ ......(ii) $c= \frac{a \space sin \space C}{sin \space A}$ ......(iii)

Case-III : Two sides a, b & angle A opposite to one of them is given

(a) If $a < b \space sin A$ No triangle exist

(b) if $a = b \space sin A \space \& \space A$ is acute, then one triangle exist which is right angled.

(c) $a > b \space sin A, a < b \space \& A$ is acute, then two triangles exist

(d) $a > b \space sin A, a > b \space \& \space A$ is acute, then one triangle exist

(e) $a > b \space sin A, \space \& \space A$ is obtuse, then there is one triangle if $a > b$ & no triangle if $a < b.$

Note: Case-III can be analyzed algebraically using the Cosine rule as $cos A = \frac{b^2+c^2-a^2}{2bc},$ which is quadratic in c.

18. Angles of Elevation and Depression:

Let OP be a horizontal line in the vertical plane in which an object R is given, and let OR be joined.

In Fig. (a), where the object R is above the horizontal line OP, the angle POR is called the angle of elevation of the object R as seen from the point O. In Fig. (b), where the object R is below the horizontal line OP, the angle POR is called the angle of depression of the object R, as seen from the point O.

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