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Learn/ 11th Class Mathematics/ Unit 11 /Numericals

Unit 11: Trigonometric Functions and their Graphs — Exercise 11 1

11th Class Mathematics · Unit 11: Trigonometric Functions and their Graphs

1.(i) Determine whether the following functions are even, odd or neither odd nor even. (i) sin²x (ii) Determine whether the following functions are even, odd or neither odd nor even. (ii) sin x + cos x (iii) Determine whether the following functions are even, odd or neither odd nor even. (iii) sin⁴x + cos⁴x (iv) Determine whether the following functions are even, odd or neither odd nor even. (iv) tan x + sec x (v) Determine whether the following functions are even, odd or neither odd nor even. (v) 1/cosec³x (vi) Determine whether the following functions are even, odd or neither odd nor even. (vi) (sin x + sin 3x)/(cos x + cos 3x)
Given
(i) Function f(x) = sin2 x
Formula
Replace x by -x f(-x) = [sin(-x)]2
Using sin(-θ) = -sin θ f(-x) = [-sin x]2 = sin2 x
Result
Property f(-x) = f(x) ⇒ even function
Given
(ii) Function f(x) = sin x + cos x
Formula
Replace x by -x f(-x) = sin(-x) + cos(-x)
Using sin(-θ) = -sin θ and cos(-θ) = cos θ f(-x) = -sin x + cos x
Result
Property f(-x) neq f(x) and f(-x) neq -f(x) ⇒ neither
Given
(iii) Function f(x) = sin4 x + cos4 x
Formula
Replace x by -x f(-x) = [sin(-x)]4 + [cos(-x)]4
Using sin(-θ) = -sin θ and cos(-θ) = cos θ f(-x) = [-sin x]4 + [cos x]4 = sin4 x + cos4 x
Result
Property f(-x) = f(x) ⇒ even function
Given
(iv) Function f(x) = tan x + sec x
Formula
Replace x by -x f(-x) = tan(-x) + sec(-x)
Using tan(-θ) = -tan θ and sec(-θ) = sec θ f(-x) = -tan x + sec x
Result
Property f(-x) neq f(x) and f(-x) neq -f(x) ⇒ neither
Given
(v) Function f(x) = 1csc3 x
Formula
Replace x by -x f(-x) = 1[csc(-x)]3
Using cosec(-θ) = -cosec θ f(-x) = 1[-csc x]3 = 1-csc3 x = -1csc3 x
Result
Property f(-x) = -f(x) ⇒ odd function
Given
(vi) Function f(x) = sin x + sin 3xcos x + cos 3x
Formula
Replace x by -x f(-x) = sin(-x) + sin(-3x)cos(-x) + cos(-3x)
Simplify f(-x) = -sin x - sin 3xcos x + cos 3x
Result
Property f(-x) = -sin x + sin 3xcos x + cos 3x = -f(x) ⇒ odd function
2.(i) Find the periods of the following functions. (i) sin 5x (ii) Find the periods of the following functions. (ii) cos 7x (iii) Find the periods of the following functions. (iii) tan 3x (iv) Find the periods of the following functions. (iv) cot(x/2) (v) Find the periods of the following functions. (v) 19sin(π/20 · x) (vi) Find the periods of the following functions. (vi) cosec(2x/5) (vii) Find the periods of the following functions. (vii) (1/2)sin(3x/2 - π/2) (viii) Find the periods of the following functions. (viii) -5 - 3sec(7πx + π/4) (ix) Find the periods of the following functions. (ix) 12 + 10tan(π/30 · x) (x) Find the periods of the following functions. (x) 6 - 4cot(7x/4 + π/4) (xi) Find the periods of the following functions. (xi) 9 + 30sec(x/15 + 2π/15)
Given
(i) Function f(x) = sin 5x
Formula
Period of sine Period = 2pi
Condition for period T sin 5(x + T) = sin 5x ⇒ sin(5x + 5T) = sin 5x
Using sin(θ + 2π) = sin θ 5T = 2pi ⇒ T = 2pi5
Given
(ii) Function f(x) = cos 7x
Formula
Period of cosine Period = 2pi
Condition for period T cos 7(x + T) = cos 7x ⇒ 7T = 2pi
Result
Period T = 2pi7
Given
(iii) Function f(x) = tan 3x
Formula
Period of tangent Period = pi
Condition for period T tan 3(x + T) = tan 3x ⇒ 3T = pi
Result
Period T = pi3
Given
(iv) Function f(x) = cotleft(x2right)
Formula
Period of cotangent Period = pi
Condition for period T cotleft(x + T2right) = cotleft(x2right) ⇒ T2 = pi
Result
Period T = 2pi
Given
(v) Function f(x) = 19sinleft(pi20xright)
Formula
Period of sine Period = 2pi
Condition for period T 19sinleft[pi20(x + T)right] = 19sinleft(pi20xright) ⇒ pi T20 = 2pi
Result
Period T = 40
Given
(vi) Function f(x) = cscleft(2x5right)
Formula
Period of cosecant Period = 2pi
Condition for period T cscleft[2(x + T)5right] = cscleft(2x5right) ⇒ 2T5 = 2pi
Result
Period T = 5pi
Given
(vii) Function f(x) = 12sinleft(3x2 - pi2right)
Formula
Period of sine Period = 2pi
Coefficient of x is 3/2 12sinleft[32(x + T) - pi2right] = 12sinleft(3x2 - pi2right) ⇒ 3T2 = 2pi
Result
Period T = 4pi3
Given
(viii) Function f(x) = -5 - 3secleft(7pi x + pi4right)
Formula
Period of secant Period = 2pi
Coefficient of x is 7π -3secleft[7pi(x + T) + pi4right] = -3secleft(7pi x + pi4right) ⇒ 7pi T = 2pi
Result
Period T = 27
Given
(ix) Function f(x) = 12 + 10tanleft(pi30xright)
Formula
Period of tangent Period = pi
Coefficient of x is π/30 10tanleft[pi30(x + T)right] = 10tanleft(pi30xright) ⇒ pi T30 = pi
Result
Period T = 30
Given
(x) Function f(x) = 6 - 4cotleft(7x4 + pi4right)
Formula
Period of cotangent Period = pi
Coefficient of x is 7/4 -4cotleft[74(x + T) + pi4right] = -4cotleft(7x4 + pi4right) ⇒ 7T4 = pi
Result
Period T = 4pi7
Given
(xi) Function f(x) = 9 + 30secleft(x15 + 2pi15right)
Formula
Period of secant Period = 2pi
Coefficient of x is 1/15 30secleft[115(x + T) + 2pi15right] = 30secleft(x15 + 2pi15right) ⇒ T15 = 2pi
Result
Period T = 30pi