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# Xi maths ch3_trigo_func_formulae

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### Xi maths ch3_trigo_func_formulae

1. 1. 1 Class XI: Maths Chapter 3: Trigonometric Functions Top Formulae 180o1. 1 radian = = 57o16 approximately π π2. 1o = radians = 0.01746 radians approximately 180o3. s= r θ Length of arc= radius × angle in radian This relation can only be used when θ is in radians π4. Radian measure= × Degree measure 180 1805. Degree measure = × Radian measure π6. Trigonometric functions in terms of sine and cosine 1 cos ec x = , x ≠ nπ, where n is any int eger sin x 1 π s ec x = , x ≠ (2n + 1) , where n is any int eger cos x 2 sin x π tan x = , x ≠ (2n + 1) , where n is any int eger cos x 2 1 cot x = , x ≠ nπ, where n is any int eger tan x7. Fundamental Trigonometric Identities sin2x + cos2x = 1 1 + tan2x = sec2 x 1 + cot2x = cosec2xGet the Power of Visual Impact on your sideLog on to www.topperlearning.com
2. 2. 28 Values of Trigonometric ratios: π π π π 3π 0° π 2π 6 4 3 2 2 1 1 3 sin 0 10 0 –1 0 2 2 2 3 1 1 cos 1 0 –1 0 1 2 2 2 1 not not tan 0 1 3 0 0 3 defined defined9. Domain and range of various trigonometric functions: Function Domain Range  π π y = sin x − 2 , 2  [–1, 1]   y = cos x 0, π   [–1, 1]  π π y = cosec x  − 2 , 2  − {0} R – (–1,1)   π y = sec x 0, π −     R – (–1, 1) 2  π π y = tan x − 2 , 2  R   y = cot x ( 0, π ) R10. Sign Convention I II III IV sin x + + – – cos x + – – + tan x + – + – cosec x + + – – sec x + – – + cot x + – + –Get the Power of Visual Impact on your sideLog on to www.topperlearning.com
3. 3. 311. Behavior of Trigonometric Functions in various Quadrants I quadrant II quadrant III quadrant IV quadrant increases from decreases from decreases from increases from sin 0 to 1 1 to 0 0 to –1 –1 to 0 decreases from decreases from increases from increases from cos 1 to 0 0 to –1 –1 to 0 0 to 1 increases from increases from increase from 0 increases from tan 0 to ∞ –∞ to 0 to –∞ –∞ to 0 decrease from decreases from decreases from decreases from cot ∞ to 0 0 to –∞ ∞ to 0 0 to –∞ increases from increase from decreases from decreases from sec 1 to ∞ –∞ to –1 –1 to –∞ ∞ to 1 decreases from increases from increases from decreases from cosec ∞ to 1 1 to ∞ –∞ to –1 –1 to –∞12. Basic Formulae (i) cos (x + y) = cos x cos y – sin x sin y (ii) cos (x - y) = cos x cos y + sin x sin y (iii) sin (x + y) = sin x cos y + cos x sin y (iv) sin (x – y) = sin x cos y – cos x sin y π If none of the angles x, y and (x + y) is an odd multiple of , then 2 tan x + tan y (v) tan (x + y) = 1 − tan x tan y tan x − tan y (vi) tan (x – y) = 1 + tan x tan y If none of the angles x, y and (x + y) is a multiple of π, then cot x cot y − 1 (vii) cot (x + y) = cot x cot y cot x cot y − 1 (viii) cot (x – y) = cot y − cot xGet the Power of Visual Impact on your sideLog on to www.topperlearning.com
4. 4. 413. Allied Angle Relations π  cos  − x  = sin x 2  π  sin  − x  = cos x  2  π  π  con  + x  = –sin x sin  + x  = cos x 2  2  cos (π – x) = –cos x sin (π – x) = sin x cos (π + x) = –cos x sin (π + x) = –sin x cos (2π – x) = cos x sin (2π – x) = –sin x cos (2nπ + x) = cos x sin (2nπ + x) = sin x14. Sum and Difference Formulae x+y x−y (i) cos x + cos y = 2 cos cos 2 2 x+y x−y (ii) cos x – cos y = −2 sin 2 sin 2 x+y x−y (iii) sin x + sin y = 2 sin cos 2 2 x+y x−y (iv) sin x – sin y = 2 cos 2 sin 2 (v) 2cos x cos y = cos (x + y) + cos (x – y) (vi) –2sin x sin y = cos (x + y) – cos (x – y) (vii) 2sin x cos y = sin (x +y) + sin (x – y) (viii) 2cos x sin y = sin (x + y) – sin (x – y)15. Multiple Angle Formulae 1 − tan2 x (i) cos 2x = cos2 x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x = 1 + tan2 x 2 tan x (ii) sin 2x = 2 sin x cos x = 1 + tan2 x 2 tan x (iii) tan 2x = 1 − tan2 x (iv) sin 3x = 3 sin x – 4 sin³ x (v) cos 3x = 4 cos³ x – 3 cos x 3 tan x − tan3 x (vi) tan 3x = 1 − 3 tan2 xGet the Power of Visual Impact on your sideLog on to www.topperlearning.com
5. 5. 516. Trigonometric Equations No. Equations General Solution Principal Value 1 sin θ = 0 θ = nπ, n∈Z θ=0 cos θ = 0 π π θ = (2n + 1) , θ= 2 2 2 n∈Z 3 tan θ = 0 θ = nπ θ=0 sin θ = sin α θ = nπ + (-1)ⁿ α θ=α 4 n∈Z 5 cos θ = cos α θ = 2nπ ± α n∈Z θ = 2α, α > 0 6 tan θ = tan α θ = nπ + α n∈Z θ=α14. (i) sin θ = k = sin (nπ + (–1)ⁿ α), n є Z θ = nπ + (–1)ⁿ α, n є Z cosec θ = cosec α ⇒ sin θ = sin α θ = nπ + (–1)ⁿ α, n є Z (ii) cos θ = k = cos (2nπ ± α), n є Z θ = 2nπ ± α, n є ZGet the Power of Visual Impact on your sideLog on to www.topperlearning.com