Classxitrigonometry Formulae

ClassXITrigonometry Formulae

● 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 is22/7; 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 radiusrby an arc of lengthsthen Θ =s/ror, s = rΘ.

● Trigonometrical Ratios of some Standard Angles:

● Trigonometrical Ratios for Associated Angles:

(ii) If Θ is a positive acute angle andnis aneveninteger 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 andnis anoddinteger 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) = sin2A - sin2B = cos2B - cos2A.
(vi) cos (A + B) cos (A - B) = cos2A - sin2B = cos2B - sin2A.
(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) 2 sin A cos B = sin (A + B) + sin(A - B).
(x) 2 cos A sin B = sin (A + B ) - sin (A - B).
(xi) 2 cos A cos B = cos (A + B ) + cos (A - B).
(xii) 2 sin A sin B = cos (A - B) - cos (A + B).
(xiii) sin C + sin D = 2 sin(C + D)/2cos(C - D)/2.
(xiv) sin C - sin D = 2 cos(C + D)/2sin(C - D)/2.
(xv) cos C + cos D = 2 cos(C + D)/2cos(C - D)/2.
(xvi) cos C - cos D = 2 sin(C + D)/2sin(D - C)/2.

● Multiple Angles:

(i) sin 2Θ = 2 sin Θ cos Θ.
(ii) cos 2Θ = cos2Θ - sin2Θ.
(iii) cos 2 Θ = 2 cos2Θ - 1.
(iv) cos 2Θ = 1 - 2 sin2Θ.
(v) 1 + cos2Θ = 2 cos2Θ.
(vi) 1 - cos2Θ = 2 sin2Θ.
(vii) tan2Θ = (1 - cos 2Θ)/(1 + cos 2Θ).
(viii) sin 2Θ = (2 tan Θ)/(1 + tan2Θ)
(ix) cos 2Θ = (1 - tan2Θ)/(1 + tan2Θ).
(x) tan 2Θ = (2 tan Θ)/(1 - tan2Θ).
(xi) sin 3Θ = 3 sin Θ - 4 sin3Θ.
(xii) cos 3ф = 4 cos3Θ - 3 cos Θ.

(xiii) tan 3Θ = (3 tan Θ - tan3Θ)/(1 - 3 tan2Θ).

● General Solutions:

(a) If sin Θ = 0 then, Θ = nπ.
(b) If sin Θ = sin α then, Θ = nπ + (-1)nα.
(c) If cos Θ = 0 then, Θ = (2n + 1)(π/2).
(d) If cos Θ = cos α then, Θ = 2nπ ± α.
(e) If tan Θ = 0 then, Θ = nπ.
(f) If tan Θ = tan α then, Θ = 2nπ + α where, n = 0 or any integer.

● Inverse Circular Functions:

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

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