Grade 12 - Exp. / Complex numbers and trigonometry 45 exercises corrected

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1. Reminders E.2304 Here is the trigonometric table of notable angles : α 0 ı 6 ı 4 ı 3 ı 2 cos α 1 √ 3 2 √ 2 2 1 2 0 sin α 0 1 2 √ 2 2 √ 3 2 1 tan α 0 √ 3 3 1 √ 3 × Determine the exact values of the expressions below : a sin 7 ı 3 b cos − 5 ı 4 c cos 5 ı 6 E.7605 Proposal: formula for associated angles cos( − x ) = cos x sin( − x ) = − sin x cos( ı + x ) = − cos x sin( ı + x ) = − sin x cos( ı − x ) = − cos x sin( ı − x ) = sin x cos ı 2 + x = − sin x sin ı 2 + x = cos x cos ı 2 − x = sin x sin ı 2 − x = cos x Let ¸ be a real number. Simplify the following expressions : a cos ı 2 + ¸ b sin ¸ +3 · ı c cos ¸ − ı 2 d sin ı 2 − ¸ E.2575 Using the relationship : tan x = sin x cos x where x = ı 2 + k · ı simplify the following expressions : a tan x + ı b tan ı 2 − x E.8577 Express the following expression us-ing the trigonometric ratios of ı 5 : A = 2 · cos 4 ı 5 + 3 · sin 6 ı 5 − 4 · sin 3 ı 10 E.2230 1 Establish equality: cos ı 6 + cos 5 ı 6 = 0 2 Determine the value of the coefficients ¸ and ˛ achieving the following equality: 2 · cos − ı 7 +3 · cos 8 ı 7 − 2 · sin 6 ı 7 +sin − ı 7 = ¸ · cos ı 7 + ˛ · sin ı 7 2. Addition formulas E.6808 In the plane provided with an orthonormal reference frame O ; I ; J , consider the points M and N such that : −−→ OM = a ; −−→ ON = b −→ OI ; −−→ OM = ¸ ; −→ OI ; −−→ ON = ˛ 1 a Determine the coordinates of points M and N . b Give an expression for the scalar product −−→ OM · −−→ ON 2 a Give the measure of the oriented angle: −−→ OM ; −−→ ON b Give another expression for −−→ OM · −−→ ON . 3 Deduce the equality: cos ˛ − ¸ = cos ˛ · cos ¸ + sin ˛ · sin ¸ E.2616 Addition and subtraction formulas cos( a + b ) = cos a · cos b − sin a · sin b cos( a − b ) = cos a · cos b + sin a · sin b sin( a + b ) = sin a · cos b + cos a · sin b sin( a − b ) = sin a · cos b − cos a · sin b Determine a simplification of the following expressions : 1 cos 2 x · cos x − sin 2 x · sin x 2 sin 3 x · cos 2 x − sin 2 x · cos 3 x E.2614 1 Noting the equality ı 12 = ı 3 − ı 4 , determine the values of cos ı 12 and sin ı 12 . 2 Determine the values of : cos 7 ı 12 ; sin 7 ı 12 E.3794 1 a Expand expression : sin x + ı 4 . b Solve the equation : sin x +cos x =1 2 a Expand expression : cos x + ı 3 . b Solve the equation : cos x − 3 · sin x =1 https://chingmath.fr chapExoCorrec/2304 sacados/2304 chapExoCorrec/7605 sacados/7605 chapExoCorrec/2575 sacados/2575 chapExoCorrec/8577 sacados/8577 chapExoCorrec/2230 sacados/2230 chapExoCorrec/6808 sacados/6808 ˛o¸oOIJMN chapExoCorrec/2616 sacados/2616 chapExoCorrec/2614 sacados/2614 chapExoCorrec/3794 sacados/3794
E.2615 Show the following relationship : sin( a + b ) · cos( a − b ) = sin a · cos a + cos b · sin b E.3089 1 Simplify the following expression : cos x + ı 4 − cos x − ı 4 2 Establish the following equality: sin 5 x sin 2 x + sin 2 x sin x = sin 3 x 2 sin(2 x ) · sin( x ) 3 Solve the following equation : 3 2 · cos(2 x ) + 1 2 · sin(2 x ) = cos ı 7 E.4722 Consider the square ABCD . Let M be a point belonging to the semicircle C of diameter [ AB ] lying outside the square ABCD . Consider the points N , P and Q such that the quadrilateral MNPQ is a square whose points A , B , C , D belong respec-tively to the straight lines ( MQ ) , ( MN ) , ( NP ) , ( PQ ) . 1 Show that the triangles AMB , ADQ , CDP and BCN are isometric. 2 Note A the area of the square ABCD , A the area of the square MNPQ and ¸ the geometric measure of the angle ∠ BAM . Show that the equality: A A =1+sin(2 ¸ ) 3 For what value of ¸ , the area A is twice the area A . 3. Duplication formulas E.2613 Duplication formulas cos(2 a ) = cos a 2 − sin a 2 cos(2 a ) = 2 · cos a 2 − 1 cos(2 a ) = 1 − 2 · sin a 2 sin(2 a ) = 2 · sin a · cos a 1 Establish the following relationship : cos ı 8 2 = 1 4 · 2 + 2 2 Deduce the value of cos ı 8 . 3 Establish the relationship : sin ı 8 = 1 2 2 − 2 4. Operation on trigonometric entries E.8578 Consider the two complex numbers : z 1 = 1 + i ; z 2 = 3 + 3 · i 1 Determine the trigonometric writing of the complex num-bers z 1 and z 2 . 2 Perform the product z 1 · z 2 . Deduce the trigonometric form of this product. 3 a Express the modulus of the product z 1 · z 2 in terms of the moduli of z 1 and z 2 . b Express the product argument z 1 · z 2 in terms of the arguments z 1 and z 2 . https://chingmath.fr chapExoCorrec/2615 sacados/2615 chapExoCorrec/3089 sacados/3089 chapExoCorrec/4722 sacados/4722 de moi verifier si correct ABCDMNPQ chapExoCorrec/2613 sacados/2613 chapExoCorrec/8578 sacados/8578
E.8579 Consider the two complex numbers : z 1 = 2 − 2 · i ; z 2 = − 1 + 3 · i 1 Determine the trigonometric writing of the complex num-bers z 1 and z 2 . 2 Give the trigonometric writing of the quotient z 1 z 2 . 3 a Express the modulus of the product z 1 z 2 in terms of the moduli of z 1 and z 2 . b Express the product argument z 1 z 2 in terms of the ar-guments of z 1 and z 2 . E.5952 Give the trigonometric form of the complex numbers : a z 1 = 2 · cos ı 4 − i · sin ı 4 b z 2 = − 3 · cos 2 ı 3 + i · sin 2 ı 3 c z 3 = cos ı 6 + i · sin − ı 6 d z 4 = 2 · cos ı 4 + i · sin 3 ı 4 E.3828 Find all pairs ( z 1 ; z 2 ) of com-plex numbers satisfying the conditions : z 1 · z 2 = 1 2 z 1 + 2 · z 2 = 3 Determine the trigonometric writing of each of the numbers thus obtained. E.3826 Consider the two complex numbers : z 1 = 6 − i · 2 2 ; z 2 = 1 − i 1 Give the trigonometric writing of the complex numbers z 1 and z 2 . 2 a Establish the following equality: cos − ı 6 + i · sin − ı 6 cos − ı 4 + i · sin − ı 4 = cos ı 12 + i · sin ı 12 b We define the number Z by the relation: Z = z 1 z 2 Determine the trigonometric writing of the complex number Z . 3 Deduce that : cos ı 12 = 6 + 2 4 ; sin ı 12 = 6 − 2 4 4 Consider the equation with real unknown x : 6 + 2 · cos x + 6 − 2 · sin x = 2 Solve this equation in R . 5. Exponential writing E.5365 Give the algebraic form of the fol-lowing complex numbers : a z 1 = 3 · e i · π b z 2 = 2 · e i · π 4 c z 3 = 2 3 · e − i · π 6 E.3848 Determine the exponential form of the following complex numbers : a z 1 = 5 b z 2 = − 3 c z 3 = − 3 · i d z 4 = − 3 + 3 · i e z 5 = − 2 3 − 2 · i f z 6 = 3 − 3 · i E.3831 1 Let z be a complex number admitting the exponential form : z = r · e i · θ où r ∈ R ∗ + , „ ∈ R Determine the exponential form of the following complex numbers : a z b − z 2 a Justify the following equality: 2 · e − i · π 3 = − 2 · e i · 2 π 3 b Deduce the exponential writing of : 2 · e − i · π 3 + 3 · e i · 2 π 3 6. Product, exponential quotient E.3798 1 Give the exponential writing of the following two com-plex numbers : z 1 =1 − i ; z 2 =1+i 2 Deduce the exponential writing of the complex number: z = 1 − i 1 + i E.8581 Determine the exponential form of the complex number z 3 defined by: z 3 = 1+i 1 −√ 3 · i E.6104 Consider the two complex numbers : z 1 = 1 + i ; z 2 = 3 + i 1 Determine the exponential writings of the numbers z 1 and z 2 . 2 We note z 3 and z 4 the two complex numbers defined by: z 3 = z 1 · z 2 ; z 4 = z 1 z 2 Determine the exponential writings of the numbers z 3 and z 4 . https://chingmath.fr chapExoCorrec/8579 sacados/8579 chapExoCorrec/5952 sacados/5952 chapExoCorrec/3828 sacados/3828 Extrait Bordeaux 1976 chapExoCorrec/3826 sacados/3826 chapExoCorrec/5365 sacados/5365 chapExoCorrec/3848 sacados/3848 chapExoCorrec/3831 sacados/3831 chapExoCorrec/3798 sacados/3798 chapExoCorrec/8581 sacados/8581 chapExoCorrec/6104 sacados/6104
E.5366 Consider the two complex numbers given below : z 1 = 3 · e i · π 3 ; z 2 = 6 · e − i · π 6 Determine a simplified expression for the following calcula-tions : a z 1 · z 2 b z 1 z 2 c z 1 + z 2 E.4253 In C , consider the following two complex numbers : z A = 1 − i ; z B = 2 + 3 + i 1 Determine the modulus and an argument of z A . 2 Write z B z A under algebraic script. 3 Show that : z B z A =(1+ 3) · e i · π 3 4 Deduce the exponential writing of z B . E.3827 Consider in C the complex numbers z 1 and z 2 of module 1 and arguments ¸ and ˛ re-spectively. Show that : z 1 + z 2 2 z 1 · z 2 is a positive or zero real. E.6380 Consider the sequence of com-plex numbers z n defined by: z 0 = 3 − i ; z n +1 = 1 + i · z n pourt tout n ∈ N 1 Determine the algebraic form of z 1 . 2 Determine the exponential form of z 0 and 1+i . Deduce the exponential form of z 1 . 3 Deduce from previous questions the exact value of cos ı 12 E.8582 Consider the complex number z de-fined by: z = 1 + i 3 + 3 · i 1 Determine the algebraic writing of the complex number z . 2 Deduce the principal measure of the angle „ achieving: cos „ = 2 + 6 4 ; sin „ = 6 − 2 4 7. Exponential writing and main argument measure E.5381 Consider the following two complex numbers given in their exponential form : z 1 = 2 · e i · 2 π 3 ; z 2 = 3 · e i · 3 π 4 Give the exponential writing of the following expressions : a z 1 · z 2 b z 1 2 · z 2 c z 2 z 1 3 Hint: The main measures of the arguments will be given E.3829 1 a Let z 1 be the complex number whose algebraic form is : z 1 = − 2 · 3+2 · i Determine the exponential writing of this number. b Let z 2 be the complex number verifying: | z 2 | = 2 ; arg( z 2 ) = 3 4 · ı Determine the algebraic writing of the complex z 2 2 Determine, at your convenience, either the algebraic or exponential writing of the following complex numbers : a z 1 · z 2 b z 1 + z 2 Hint: The main measures of the arguments will be given E.3830 1 We define the two complex numbers z and z by: z = 3 − 4 · i ; z = − 2 + i Determine the algebraic writing of the following num-bers : a z + z b z · z c z · z + i 2 Let us consider the two complex numbers z and z ad-mitting as algebraic writing: z = 2 · e i · π 4 ; z = e i · 5 π 6 Determine the exponential writing of the following com-plex numbers : a − z b z c z · z d z z Hint: The main measures of the arguments will be given E.6081 The complex plane is referred to a O ; −→ u ; −→ v direct orthonormal coordinate system. Consider the application z of C in C defined by: f : z ↦−→ z 2 . Consider the complex: a = 2 − i · 2 1 Express a in exponential form. 2 Deduce the complex numbers antecedent to the number a by f . 8. Exponential writing power E.8583 Consider the complex number z de-fined by: z =1+i Establish that the number z 100 is a real number. https://chingmath.fr chapExoCorrec/5366 sacados/5366 chapExoCorrec/4253 sacados/4253 Extrait de Liban Juin 2011 chapExoCorrec/3827 sacados/3827 Extrait Montpellier 1982 chapExoCorrec/6380 sacados/6380 Extrait de Liban Mai 2014 chapExoCorrec/8582 sacados/8582 chapExoCorrec/5381 sacados/5381 chapExoCorrec/3829 sacados/3829 chapExoCorrec/3830 sacados/3830 chapExoCorrec/6081 sacados/6081 chapExoCorrec/8583 sacados/8583
E.4325 For each of the following propositions, indicate whether it is true or false and give a demonstration of the chosen answer. 1 So z =3+i · 3 . Proposition 1: For any non-zero natural number n , z 3 · n is pure imaginary. 2 Let z be a non-zero complex number. Proposition 2: If ı 2 is an argument of z then : ⏐ ⏐ i + z ⏐ ⏐ = 1 + ⏐ ⏐ z ⏐ ⏐ 3 Let z be a non-zero complex number. Proposition 3: If the modulus of z is equal to 1 then z 2 + 1 z 2 is a real number. E.5325 Consider the complex number: a = − 3+ i 2011 . Indicate whether the following statement is true or false, jus-tifying the answer: ˇ The complex number a is a pure imaginary number. ı 9. Euler formula E.8584 Definition: on linearizes an expression of the form cos p ( x ) · sin q ( x ) , where p;q ∈ N when transformed into a linear combination of expressions of the form cos( n · x ) and sin( m · x ) , where n;m ∈ N Establish the following linearization: cos( x ) 2 · sin( x ) 2 = 1 8 − 1 8 · cos(4 x ) E.5947 We wish to linearize the expression cos 3 ( x ) : 1 Using Euler’s formula and the binomial formula, estab-lish : cos 3 ( x ) = e − 3i · x + 3 · e − i · x + 3 · e i · x + e 3i · x 8 2 Deduce the identity: cos 3 ( x ) = cos(3 x ) + 3 · cos( x ) 4 E.599 For any real number x and for any non-zero integer k , establish the relation: cos( x ) 2 k = 1 4 k · 2 k k + 1 2 2 k − 1 · k − 1 =0 2 k ‘ · cos (2 k − 2 ‘ ) x 10. Moivre formula E.8585 Prerequisite: For any natural number n and any real number „ , we have : cos „ + i · sin „ n = cos n · „ + i · sin n · „ 1 Using the binomial formula, establish the identity: cos „ +i · sin „ 3 = 4 · cos( „ ) 3 − 3 · cos „ +i 3 · sin „ − 4 · sin( „ ) 3 2 Using Moivre’s formula, establish the identities: cos(3 „ )=4 · cos( „ ) 3 − 3 · cos „ ; sin(3 „ )=3 · sin „ − 4 · sin( „ ) 3 11. Courses E.3299 Reminders: If z is a non-zero complex number, we have the follow-ing equivalence : | z | = r arg z = „ up to 2 ı ⇐⇒ z = r cos „ + i sin „ r > 0 For all real numbers a and b : cos a + b = cos a · cos b − sin a · sin b sin a + b = sin a · cos b + sin b · cos a Let z 1 and z 2 be two nonzero complex numbers. Prove the relations : | z 1 z 2 | = | z 1 | · | z 2 | arg( z 1 z 2 ) = arg ( z 1 ) + arg ( z 2 ) to within 2 ı . E.3301 Recall that for any non-zero vec-tor −→ w , of affix z , we have : | z | = −→ w ; arg( z ) = −→ u ; −→ w à 2 ı près. Prerequisite: we know that if z and z are two non-zero complex numbers, then : arg( z · z ) = arg( z ) + arg ( z ) Let z and z be two non-zero complex numbers. Show that : arg z z = arg( z ) − arg( z ) https://chingmath.fr chapExoCorrec/4325 sacados/4325 Extrait de Polynesie Juin 2011 chapExoCorrec/5325 sacados/5325 chapExoCorrec/8584 sacados/8584 chapExoCorrec/5947 sacados/5947 chapExoCorrec/599 sacados/599 chapExoCorrec/8585 sacados/8585 chapExoCorrec/3299 sacados/3299 chapExoCorrec/3301 sacados/3301 Extrait d'Asie - Juin 2006
E.5951 Consider two non-zero complex num-bers z and z with respective modules r and r and respective arguments „ and „ . 1 Give the trigonometric writing of the complex numbers z and z . 2 Show that the complex number z · z is a complex number with modulus ( r · r ) and argument ( „ + „ ) . 3 Show that the complex number z z is a complex number with module r r and argument ( „ − „ ) . 12. Unclassified financial years E.5280 Let the complex numbers be : z 1 = 2 + i · 6 ; z 2 = 2 + 2 · i ; Z = z 1 z 2 1 Write Z in algebraic form. 2 Give the modules and arguments of z 1 , z 2 and Z . 3 Deduct cos ı 12 and sin ı 12 . E.5281 Consider the following equation ( E ) : z 2 +2cos ı 5 · z +1=0 Indicate whether the following proposition is true or false, justifying your assertion : ˇ The equation ( E ) has two complex solutions with moduli equal to 1 . ı https://chingmath.fr chapExoCorrec/5951 sacados/5951 chapExoCorrec/5280 sacados/5280 chapExoCorrec/5281 sacados/5281 Extrait du Liban Juin 2008