Simple Math Formula on Trigonometry

Simple math formula on trigonometry is given in such an order that students can easily get the formula.


Trigonometry

● Measurement of Trigonometrical Angles:

(i) The angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle is called a radian.

(ii) A radian is a constant angle. 

One radian = (2/π) rt. angle = 57°17’44.8” (approx.) 

(iii) 1 rt. angle = 90° ; 1° = 60’ ; 1‘ = 60”. 

(iv) 1 rt. angle = 100ᵍ ; 1ᵍ = 100’ ; 1‵ = 100‶.

(v) πᶜ 180° = 200ᵍ.

(vi) The circumference of a circle of radius r is 2πr where π is a constant; approximate value of π is ²²/₇; more accurate value of π is 3.14159 (approx.).

(vii) If Θ be the radian measure of an angle subtended at the centre of a circle of radius r by an arc of length s then Θ = ˢ/₀ or, s = rΘ.


● Trigonometrical Ratios of some Standard Angles:

Trigonometrical Ratios of some Standard Angles

● Trigonometrical Ratios for Associated Angles:

Trigonometrical Ratios for Associated Angles

(ii) If Θ is a positive acute angle and n is an even integer then,

(a) sin (n ∙ 90° ± Θ) = sin Θ or, (- sin Θ)

(b) cos (n ∙ 90° ± Θ) = cos Θ or, (- cos Θ)

(c) tan (n ∙ 90° ± Θ) = tan Θ or, (- tan Θ).

(iii) If Θ is a positive acute angle and n is an odd integer then,

(a) sin (n ∙ 90° ± Θ) = cos Θ or, (- cos Θ)

(b) cos (n ∙ 90° ± Θ) = sin Θ or, (- sin Θ)

(c) tan (n ∙ 90° ± Θ) = cot ф or (- cot Θ).



● Compound Angles:

(i) sin (A + B) = sin A cos B + cos A sin B.

(ii) sin ( A - B) = sin A cos B - cos A sin B.

(iii) cos (A + B) = cos A cos B + sin A sin B.

(iv) cos (A - B) = cos A cos B + sin A sin B.

(v) sin (A + B) sin (A - B) = sin² A - sin² B = cos² B - cos² A.

(vi) cos (A + B) cos (A - B) = cos² A - sin² B = cos² B - sin² A.

(vii) tan (A+ B) = (tan A + tan B)/(1 - tan A tan B).

(viii) tan (A - B) = (tan A - tan B)/(1 + tan A tan B).

(ix) cot (A + B) = (cot A cot B - 1)/(cot B + cot A).

(x) cot (A - B) = (cot A cot B + 1)/(cot B - cot A).

(xi) tan (A + B + C) = {(tan A + tan B + tan C) - (tan A tan B tan C)}/(1 - tan A tan B - tan B tan C - tan C tan A).

(xii) 2 sin A cos B = sin (A + B) + sin(A - B).

(xiii) 2 cos A sin B = sin (A + B ) - sin (A - B).

(xiv) 2 cos A cos B = cos (A + B ) + cos (A - B).

(xv) 2 sin A sin B = cos (A - B) - cos (A + B).

(xvi) sin C + sin D = 2 sin (C + D)/2 cos (C - D)/2.

(xvii) sin C - sin D = 2 cos (C + D)/2 sin (C - D)/2.

(xviii) cos C + cos D = 2 cos (C + D)/2 cos (C - D)/2.

(xix) cos C - cos D = 2 sin (C + D)/2 sin (C - D)/2.

● Multiple Angles:

(i) sin 2Θ = 2 sin Θ cos Θ.

(ii) cos 2Θ = cos² Θ - sin² Θ.

(iii) cos 2 Θ = 2 cos² Θ - 1.

(iv) cos 2Θ = 1 - 2 sin² Θ.

(v) 1 - cos2Θ = 2 cos² Θ.

(vi) 1 - cos2Θ = 2 sin² Θ.

(vii) tan² Θ = (1 - cos 2Θ)/(1 + cos 2Θ).

(viii) sin 2Θ = (2 tan Θ)/(1 + tan² Θ)

(ix) cos 2Θ = (1 - tan² Θ)/(1 + tan² Θ).

(x) tan 2Θ = (2 tan Θ)/(1 - tan² Θ).

(xi) sin 3Θ = 3 sin Θ - 4 sin³ Θ.

(xii) cos 3ф = 4 cos³ Θ - 3 cos Θ.

(xiii) tan 3Θ = (3 tan Θ - tan³ Θ)/(1 - 3 tan² Θ).

● Submultiple Angles:

(i) sin Θ = 2 sin (Θ/2) cos (Θ/2).

(ii) cos Θ = cos² (Θ/2) - sin² (Θ/2).

(iii) cos Θ = 2 cos² (Θ/2) - 1.

(iv) cos ф = 1 - 2 sin² (Θ/2).

(v) 1 + cos Θ = 2 cos² (Θ/2).

(vi) 1 - cos Θ = 2 sin² (Θ/2).

(vii) tan² (Θ/2) = (1 - cos Θ)/(1 + cos Θ).

(viii) sin Θ = [2 tan (Θ/2)]/[1 + tan² (Θ/2)].

(ix) cos Θ = [1 - tan² (Θ/2)]/[1 + tan² (Θ/2)].

(x) tan Θ = [2 tan (Θ/2)]/[1 - tan² (Θ/2)].

(xi) sin Θ = 3 sin (Θ/3) - 4 sin³ (Θ/3).

(xii) cos Θ = 4 cos³ (Θ/3) - 3 cos (Θ/2).

(xiii) (a) sin 15° = cos 75° = (√3 - 1)/(2√2).

(b) cos 15° = sin 75° = (√3 + 1)/(2√2).

(c) tan 15° = 2 - √3.

(d) sin 22 ½° = √(2 - √2).

(e) cos 22 ½° = ½ [√(2 + √2)].

(f) tan 22 ½° = √2 - 1.

(g) sin 18 ° = (√5 - 1)/4 = cos 72°.

(h) cos 36° = cos 72° = (√5 + 1)/4.

(i) cos 18° = sin 72° = ¼ [√(10 + 2√5)].

(j) sin 36° = cos 54° = ¼ [√(10 - 2√5)].



● General Solutions:

(i) (a) If sin Θ = 0 then, Θ = nπ.

(b) If sin Θ = 1 then, Θ = (4n + 1)(π/2).

(c) If sin ф = -1 then, Θ = (4n - 1)(π/2).

(d) If sin Θ = sin α then, Θ = nπ + (-1)ⁿ α.

(ii) (a) If cos Θ = 0 then, Θ = (2n + 1)(π/2).

(b) If cos Θ = 1 then, Θ = 2nπ.

(c) If cos Θ = -1 then, Θ = (2n + 1)π.

(d) If cos Θ = cos α then, Θ = 2nπ ± α.

(ii) (a) If tan Θ = 0 then, Θ = nπ.

(b) If tan Θ = tan α then, Θ = 2nπ + α where, n = 0 or any integer.



● Inverse Circular Functions:

(i) sin (sin-1 x) = x ; cos (cos-1 x) = x ; tan (tan-1 x) = x.

(ii) sin-1 (sin Θ) = Θ ; cos-1 (cos Θ) = Θ ; tan-1 (tan Θ) = Θ.

(iii) sin-1 x = cosec-1 (1/x) = cos-1 [√(1 - x2)] = sec-1 [1/√(1 - x2)]

= tan-1 [x/√(1 - x2)] = cot-1 [√(1 - x2)/x].

(iv) sin-1 x + cos-1 x = π/2 ; sec-1 x + cosec-1 x = π/2 ;

tan-1 x + cot-1 x = π/2.

(v) (a) tan-1 x + tan-1 y = tan-1 [(x + y)/(1 - xy)]

(b) tan-1 x - tan-1 y = tan-1 [(x - y)/(1 + xy)]

(vi) (a) sin-1 x + sin-1 y = sin-1 {x√(1 - y2) + y√(1 - x2)}

(b) sin-1 x - sin-1 y = sin-1 {x√(1 - y2 ) - y√(1 - x2)}

(vii) (a) cos-1 x + cos-1 y = cos-1 {xy - √(1 - x2) (1 - y2)}

(b) cos-1 x - cos-1 y = cos-1 {xy + √(1 - x2) (1 - y2)}.

(viii) 2 tan-1 x = sin-1 [2x/(1 + x2)] = cos-1 [(1 - x2)/(1 - x2)]

= tan-1 [2x/(1 - x2)].

(ix) tan-1 x + tan-1 y + tan-1 z = tan-1 [(x + y + z - xyz)/(1 - xy - yz - zx)]

(x) sin-1 x and cos-1 x are defined when -1 ≤ x ≤ 1 ; sec-1 x and cosec-1 x are defined when Ι x Ι ≥ 1 ; tan-1 x and cot-1 x are defined
when - ∞ < x < ∞.

(xi) If principal values of sin-1 x, cos-1 x and tan-1 x be α, β and γ respectively, then -π/2 ≤ α ≤ π/2, 0 ≤ β ≤ π and -π/2 ≤ γ ≤ π/2.

● Properties of Triangle:

(i) a/(sin A) = b/(sin B) = c/(sin C) = 2R.

(ii) a = b cos C + c cos B ; b = c cos A + a cos C ; c = a cos B + b cos A.

(iii) cos A = (b² + c² - a²)/2bc ; cos B = (c² + a² - b²)/2ca ;

cos C = (a² + b² - c²)/2ab

(iv) tan A = [(abc)/R] ∙[ 1/(b² + c² - a²)]

tan B = [(abc)/R] ∙ [1/(c² + a² - b²)]

tan C = [(abc)/R] ∙ [1/(a² + b² - c²)].

(v) sin (A/2) = √[(s - b) (s - c)/(bc)].

sin B/2 = √[(s - c) (s - a)/(ca)].

sin C/2 = √[(s - a) (s - b)/(ab)].

cos A/2 = √[s (s - a)/(bc)].

sin B/2 = √[s (s - b)/(ca)].

cos C/2 = √[s (s - c)/(ab)].

tan A/2 = √[(s - b) (s - c)/{s(s - c)}].

tan B/2 = √[(s - c) (s - a)/{s(s - b)}].

tan C/2 = √[(s - a) (s - b)/{s(s - c)}].

(vi) tan [(B - C)/2] = [(b - c)/(b + c)] cot (A/2).

tan [(C - A)/2] = [(c - a)/(c + a)] cot (B/2).

tan [(A - B)/2] = [(a - b)/(a + b)] cot (C/2).

(vii) ∆ = ½ [bc sin A] = ½ [ca sin B] = ½ [ab sin C].

(viii) ∆ = √{s(s - a)(s - b)(s - c)}.

(ix) R = ᵃᵇᶜ/₄₀.

(x) tan (A/2) = {(s - b)(s - c)}/∆.

tan (B/2) = {(s - c)(s - a)}/∆.

tan (C/2) = {(s - a)(s - b)}/∆

(xi) cot A/2 = {s(s - a)}/∆.

cot (B/2) = {s(s - b)}/∆.

cot (C/2) = {s(s - c)}/∆.

(xii) sin A = 2∆/bc ; sin B = 2∆/ca ; sin C = 2∆/ab

(xiii) r = ∆/s.

(xiv) r = 4R sin (A/2) sin (B/2) sin (C/2).

(xv) r = (s - a) tan (A/2) = (s - b) tan (B/2) = (s - c) tan (C/2).

(xvi) r₁ = ∆/(s - a) ; r₂ = ∆/(s - b); r₃ = ∆/(s - c) .

(xvii) r₁ = 4 R sin (A/2) cos (B/2) cos (C/2).

(xviii) r₂ = 4R sin (B/2) cos (C/2) cos (A/2).

(xix) r₃ = 4 R sin (C/2) cos (A/2) cos (B/2).

(xx) r₁ = s tan (A/2) ; r₂ = s tan (B/2) ; r₃ = s tan (C/2).

Formula




11 and 12 Grade Math

From Simple Math Formula on Trigonometry to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Recent Articles

  1. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More

  2. Construction of a Circle | Working Rules | Step-by-step Explanation |

    Jul 09, 25 01:29 AM

    Parts of a Circle
    Construction of a Circle when the length of its Radius is given. Working Rules | Step I: Open the compass such that its pointer be put on initial point (i.e. O) of ruler / scale and the pencil-end be…

    Read More

  3. Combination of Addition and Subtraction | Mixed Addition & Subtraction

    Jul 08, 25 02:32 PM

    Add and Sub
    We will discuss here about the combination of addition and subtraction. The rules which can be used to solve the sums involving addition (+) and subtraction (-) together are: I: First add

    Read More

  4. Addition & Subtraction Together |Combination of addition & subtraction

    Jul 08, 25 02:23 PM

    Addition and Subtraction Together Problem
    We will solve the different types of problems involving addition and subtraction together. To show the problem involving both addition and subtraction, we first group all the numbers with ‘+’ and…

    Read More

  5. 5th Grade Circle | Radius, Interior and Exterior of a Circle|Worksheet

    Jul 08, 25 09:55 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More