Grade 12 / Trigonometric functions 48 exercises (including 46 corrected)

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1. Reminders E.5269 Using the formulas of cos and sin of the associated angles, express in terms of cos x or sin x the following numbers : a sin 3 ı + x b cos 5 ı 2 − x c cos x − ı 2 d cos ı 2 + x e sin ı − x + cos ı 2 − x f 3 · sin ı + x − 2 · sin ı − x + 4 · sin x − ı E.5267 Simplify the argument of each of the following expressions : a tan x + ı b tan ı 2 − x c cos x − ı d sin x − ı 2 e sin x + ı 2 f cos x + ı 2 E.5268 1 Show that : cos ı 6 + cos 5 ı 6 = 0 2 Simplify the following number as far as possible. (we’ll express the result using cos ı 7 and sin ı 7 ) . 2 · cos − ı 7 + 3 · cos 8 ı 7 − 2 · sin 6 ı 7 + sin − ı 7 E.5273 1 Noting that : 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.5274 Show the following relationship : sin( a + b ) · cos( a − b ) = sin a · cos a + cos b · sin b E.5275 Determine a simplification of the follow-ing 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.5272 1 In each case, draw a trigonometric circle and represent each of the following sets : a cos x ⏐ ⏐ x ∈ ı 6 ; 2 ı 3 b sin x ⏐ ⏐ x ∈ 2 ı 3 ; 7 ı 3 2 Using a trigonometric circle, give without justification the set of solutions of the following inequations in the interval − ı ; ı : a sin x 3 2 b cos x − 1 2 c cos x < 0 E.3345 Solve, in − ı ; ı , the equations below : a cos x = cos ı 4 b sin x = sin − ı 6 c cos 2 x = cos ı 4 d cos x = cos x + ı 4 E.5270 1 Solve in the set − ı ; ı of principal measurements, the following equations : a cos x = 2 2 b sin x = − 1 2 c sin x = 3 2 d tan x = 3 3 2 Solve the following equations in R : a cos x = √ 3 2 b tan x = − 1 E.5271 Solve, in the interval − ı ; ı the fol-lowing equations : a 2 cos 2 x = 1 b sin 3 x = 3 2 c cos 2 x = cos x d sin 3 x = cos x E.5276 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, in the interval − ı ; ı the following equation : 3 2 cos(2 x ) + 1 2 sin(2 x ) = cos ı 7 2. Properties of sine and cosine functions E.2556 1 By choosing a suitable frame, show that the sequence u n n ∈ N converges to 0: u n = ( − 1) n 3 n + 1 2 By choosing a suitable frame, show that the sequence v n n ∈ N converges ; specify its limit: u n = 2 n 2 + cos n 3 n 2 + 5 E.2568 Let u n n ∈ N ∗ be a sequence defined by: u n = cos n − 2 n n Determine the limit of the sequence u n . https://chingmath.fr chapExoCorrec/5269 sacados/5269 chapExoCorrec/5267 sacados/5267 chapExoCorrec/5268 sacados/5268 chapExoCorrec/5273 sacados/5273 chapExoCorrec/5274 sacados/5274 chapExoCorrec/5275 sacados/5275 chapExoCorrec/5272 sacados/5272 chapExoCorrec/3345 sacados/3345 chapExoCorrec/5270 sacados/5270 chapExoCorrec/5271 sacados/5271 chapExoCorrec/5276 sacados/5276 chapExoCorrec/2556 sacados/2556 chapExoCorrec/2568 sacados/2568
−2·ı−ı0ı2·ı-2-112C −2·ı−ı0ı2·ı-2-112C -10-8-6-4-2246810I-4-224JO 24323436646668I-4-224JO −4·ı−3·ı−2·ı−ı0ı2·ı3·ı4·ı-112 E.3302 Consider the following two functions f and g defined on R by: f ( x ) = 2 · cos x ; g ( x ) = cos 2 · x Associate with each of these functions its representative curve shown below : E.5040 Let a n n ∈ N be a non-constant sequence of real numbers. For any integer n , we set : u n = sin( a n ) . Indicate whether the following statement is true or false and justify your answer: Statement : ˇWe can choose the sequence a n n ∈ N such that the sequence u n n ∈ N con-verges to 2 2 .ı 3. Periodicity and parity E.2920 In the ( O ; I ; J ) orthonormal, we rep-resent below the curve C f representative of the function f defined on R : The function f is periodic with period T . 1 a Give the coordinates of a vector defining a transla-tion by which the curve C f is invariant. b Determine the value of T . 2 Give the image, by the function f , of the following num-bers : a 14 b − 16 c 56 e 58 E.2922 Consider the periodic function f of pe-riod 12. Below are given some parts of the curve C represen-tative of the function f : 1 Reconstruct the plot of C f on the interval 0 ; 12 . 2 Draw up the table of variations of the function f on the interval 38 ; 50 . E.2921 For each question, show that the func-tion f admits T for period : a f ( x ) = sin 6 x − 3 ; T = ı 3 b f ( x ) = tan 2 x + ı 3 ; T = ı c f ( x ) = cos x 2 − sin x 2 ; T = ı d f ( x ) = ⏐ ⏐ ⏐ ⏐ cos 2 x + ı 3 ⏐ ⏐ ⏐ ⏐ ; T = ı 2 E.3346 Consider the function f defined on R whose image of x is defined by the relation: f ( x ) = cos x + cos( x ) 2 We call C f the representative curve of the function f . 1 Study the parity of the f function. 2 Study the periodicity of the function f . 3 a By studying the following number images : 0 ; ı 3 ; ı 2 ; 2 ı 3 ; ı ; 4 ı 3 ; 3 ı 2 ; 5 ı 3 and assuming the following properties : f is strictly decreasing on 0 ; 2 ı 3 ; f is strictly increasing on 2 ı 3 ; ı ; C f admits horizontal tangents at abscissa points : 0 ; 2 ı 3 ; ı ; − 4 ı 3 plot the curve C f on 0 ; + ∞ . b Extend this plot to the entire graph. https://chingmath.fr chapExoCorrec/3302 sacados/3302 −2·ı−ı0ı2·ı-2-112C −2·ı−ı0ı2·ı-2-112C chapExoCorrec/5040 sacados/5040 chapExoCorrec/2920 sacados/2920 -10-8-6-4-2246810I-4-224JO chapExoCorrec/2922 sacados/2922 24323436646668I-4-224JO chapExoCorrec/2921 sacados/2921 chapExoCorrec/3346 sacados/3346 −4·ı−3·ı−2·ı−ı0ı2·ı3·ı4·ı-112
4. Derivative E.5277 Determine the expression of the deriva-tives of the following functions : a f : x ↦−→ x 2 + cos x b g : x ↦−→ sin(2 x ) c h : x ↦−→ cos x · sin x d j : x ↦−→ sin x 2 E.2878 Determine the expression of the deriva-tive functions associated with the following functions : a f ( x ) = x 2 · cos x b g ( x ) = 3 x 2 − 2 · sin x 2 c h ( x ) = 3 2 · cos x d j ( x ) = cos x 3 x + sin x E.2907 Determine the expression of the derivative functions of the functions f , g , h defined below : a f ( x ) = sin x cos x b g ( x ) = (5 x − 3) 3 · cos x c h ( x ) = cos 5 x + ı 3 x 2 E.2879 Determine the derivative functions of the following functions : 5. Derivative numbers and limits E.3532 1 Determine the expression of the derivative functions of the following functions : a f ( x ) = cos x 2 b g ( x ) = sin x + cos x c h ( x ) = tan( x 2 + x ) d j ( x ) = cos x sin x 2 Determine the following limits: a lim x ↦→ 0 cos x 2 − 1 x b lim x ↦→− π 4 sin x + cos x x + ı 4 c lim x ↦→ 0 tan( x 2 + x ) x d lim x ↦→ π 2 cos x x − ı 2 · sin x E.3568 Consider the function g defined on R by: g ( x )=sin x 1 Give the derivative number of the function g in 0 . 2 Deduce the value of the limit: lim x ↦→ 0 sin x x 6. Study of functions E.3540 1 Consider the function f defined on R by: f ( x ) = cos x − 1 x for x =0 f (0) = 0 Show that the function f is continuous in 0 . 2 Consider the function g defined by the relation on R by: g ( x ) = E ( x ) · sin ı · x a Simplify the writing of the function g on each of the following intervals : − 1 ; 0 ; 0 ; 1 ; 1 ; 2 b Justify that the function g is continuous in 0 and in 1 . c Make a conjecture about the set of continuity of the function f ? d Draw the representative curve of this function on your calculator. E.3582 Consider the two sequences of real num-bers, u n n ∈ N ∗ and v n n ∈ N ∗ defined by: u n = sin 1 n 2 + sin 2 n 2 + · · · + sin n n 2 v n = 1 n 2 + 2 n 2 + · · · + n n 2 1 Show that the sequence v converges to 1 2 . 2 a Show that each of the three numerical functions of the real variable: x ↦−→ x − sin x ; x ↦−→ − 1 + x 2 2 + cos x x ↦−→ − x + x 3 6 + sin x takes only positive or zero values on the interval 0 ; + ∞ . We can use the variations of each of these three func-tions. b Justify that for any n 1 : 1 3 + 2 3 + · · · + n 3 n 4 . Deduce from a the inequality: v n − 1 6 × 1 n 2 u n v n for any n ∈ N . c Demonstrate that the sequence u is convergent ; what is its limit? https://chingmath.fr chapExoCorrec/5277 sacados/5277 chapExoCorrec/2878 sacados/2878 chapExoCorrec/2907 sacados/2907 chapExoCorrec/2879 sacados/2879 chapExoCorrec/3532 sacados/3532 chapExoCorrec/3568 sacados/3568 chapExoCorrec/3540 sacados/3540 chapExoCorrec/3582 sacados/3582
E.3559 1 We denote by g the numerical function defined on 0 ; ı by: g ( x ) = x · cos x − sin x Study g and draw up its table of variation. Deduce the sign of g ( x ) on 0 ; ı . 2 Let f be the numerical function of the real variable x defined on 0 ; ı by: f (0) = 1 f ( x ) = sin x x pour x ∈ 0 ; ı Recall that : lim x ↦→ 0 sin x x = 1 . Study the variations of f on 0 ; ı . 3 Study of f in 0 . a Prove that, for any real number x ∈ 0 ; ı : 0 x − sin x x 3 6 (For this, we will introduce the function ’ defined on 0 ; ı by : ’ ( x ) = sin x − x + x 3 6 We will calculate the derivatives ’ , ’  and ’  and deduce the sign of ’ ) b Prove that f is derivable in 0 and calculate f (0) . 4 Construct the representative curve C of the function f in an orthonormal frame O ; −→ i ; −→ j . (We took 3 cm for unit) . E.3648 We denote by g the function de-fined on − 1 ; 1 by: g (0) = 0 ; g ( x ) = 1 √ 1 − x 2 où g denotes the derivative of the function g on − 1 ; 1 ; no attempt will be made to explain g ( x ) . Consider the composite function h defined on − ı ; 0 by: h ( x ) = g cos x 1 Demonstrate that for any x of − ı ; 0 , we have h ( x )=1 , où h denotes the derivative of h . 2 Calculate h − ı 2 then give the expression for h ( x ) . 7. Study of functions and the intermediate value theorem E.6822 Specify whether the following statement is true or false, justifying the answer. Assertion The equation x − cos x =0 admits a single solution in the inter-val 0 ; ı 2 . 8. Primitive and integral E.5312 Determine a primitive of each of the following functions : a f ( x ) = cos x + sin x b g ( x ) = sin(3 x ) c h ( x ) = sin x · cos x d j ( x ) = 3 x · cos x 2 E.3932 Determine a primitive for each of the following functions : a f ( x ) = 5 x b g ( x ) = e x +1 + 1 c h ( x ) = 2 · x · e x 2 +1 d j ( x ) = cos x e k ( x ) = sin(3 x ) f ‘ ( x ) = sin x 2 E.5232 Consider the following two integrals : I = π 0 cos x 2 d x ; J = π 0 sin x 2 d x 1 Calculate: I + J ; I − J . 2 Deduce the values of I and J . E.5282 Consider the sequence x n de-fined for any non-zero natural number n by: x n = 1 0 t n · cos t d t 1 a Show that the sequence x n has positive terms. b Study the variations of the sequence x n . c What can we deduce about the convergence of the se-quence x n ? 2 a Demonstrate that, for any non-zero natural number n : x n 1 n + 1 b Deduce the limit of the sequence x n . E.5208 Calculate the integral: π 2 π 3 sin x 2 d x https://chingmath.fr chapExoCorrec/3559 sacados/3559 Algerie 1992 chapExoCorrec/3648 sacados/3648 Extrait de Metropole et Reunion Septembre 2007 chapExoCorrec/6822 sacados/6822 chapExoCorrec/5312 sacados/5312 chapExoCorrec/3932 sacados/3932 chapExoCorrec/5232 sacados/5232 chapExoCorrec/5282 sacados/5282 chapExoCorrec/5208 sacados/5208
aOCf−π2π2A 0246810121416246ijABCfCg E.5307 Consider the function f defined on 0 ; ı by: f ( x ) = cos 3 x · cos x 3 In an orthonormal frame of reference O ; I ; J (we’ll take 3 cm as the unit) , we consider the curve C f representative of the function f . 1 a Show that the function f derived from the function f has the expression : f ( x ) = − 3 · cos x 2 · sin 4 x b Draw up the sign table for the function f . c Draw up the table of variations of the function f . 2 a Show that, whatever the real x belonging to 0 ; ı , we have : f ( x ) = 1 8 · cos 6 x + 3 8 · cos 4 x + 3 8 · cos 2 x + 1 8 (Among others, we will use the formula: 2 · cos x · cos y = cos( x + y ) + cos( x − y ) ) b Calculate, in cm 2 , the area of the set E bounded by the curve C f , the x-axis and the straight lines of equations x =0 and x = ı 6 . E.6820 A factory produces bottled min-eral water. The sales department has adopted the shape of the bottle la-bels shown below in an orthonormal plane. The shape of these labels is delimited by the x-axis and the curve C with equation y = a · cos x with x ∈ − ı 2 ; ı 2 and a a strictly positive real. A disk located inside is intended to receive the information given to buyers. Consider the disk with center point A of co-ordinates 0 ; a 2 and radius a 2 . It will be assumed that this disk lies entirely below the curve C for values of a less than 1.4 . 1 Justify that the area of the domain between the x-axis, the straight lines of equation x = − ı 2 and x = ı 2 , and the curve C is equal to 2 · a unit area. 2 For aesthetic reasons, we want the area of the disk to be equal to the area of the shaded surface. What value should be given to the real a to meet this constraint? E.6821 Consider the functions f and g defined on the interval 0 ; 16 by: f ( x ) = ln x +1 ; g ( x ) = ln x +1 + 1 − cos( x ) In a reference frame O ; −→ i ; −→ j , note C f and C g the repre-sentative curves of the functions f and g . These curves are given below : Compare the areas of the two hatched surfaces on this graph. 9. Annales E.5283 The plane is referenced to an orthogonal coordinate system O ; −→ i ; −→ j . Let f be the func-tion defined on 0 ; + ∞ by: f ( x ) = e − x · cos(4 x ) and Γ its representative curve plotted in the coordinate sys-tem O ; −→ i ; −→ j at the end of the exercise. This graph will be completed as the exercise progresses. We also consider the function g defined on 0 ; + ∞ by g ( x )= e − x and we call C its representative curve in the coordinate system O ; −→ i ; −→ j . 1 a Show that, for any real number x belonging to the interval 0 ; + ∞ : − e − x f ( x ) e − x b Deduce the limit of f at + ∞ . 2 Determine the coordinates of the points common to curves Γ and C . 3 We define the sequence u n on N by: u n = f n · ı 2 . a Show that the sequence u n is a geometric sequence. Specify the reason. b Deduce the direction of variation of the sequence u n and study its convergence. 4 a Show that, for any real number x belonging to the interval 0 ; + ∞ : f ( x ) = − e − x · cos(4 x ) + 4 · sin(4 x ) b Deduce that the curves Γ and C have the same tangent at each of their common points. 5 Give an approximate value, rounded up to 10 − 1 , of the slope of the line T tangent to the curve Γ at the point with abscissa ı 2 . Complete the given graph by plotting T and C . https://chingmath.fr chapExoCorrec/5307 sacados/5307 chapExoCorrec/6820 sacados/6820 aOCf−π2π2A chapExoCorrec/6821 sacados/6821 0246810121416246ijABCfCg chapExoCorrec/5283 sacados/5283
01234-11ij ABETMTerrain vu du dessusLimiteduterrainLignemédianex 12√22√3212√22√32-12-√22-√32-12-√22-√32 E.6823 In a rugby match, a player must convert a try that has been scored at the point E (see figure opposite) located outside the segment [ AB ] . The transformation consists of kicking the ball from a point T that the player has the right to choose any où on the segment [ EM ] perpendicular to the line ( AB ) except in E . The transformation is successful if the ball passes between the posts marked by points A and B on the figure. To maximize his chances of success, the player tries to deter-mine the position of the point T that makes the angle ∠ ATB as large as possible. The aim of this exercise is therefore to investigate whether there is a position of the point T on the segment [ EM ] for which the angle ∠ ATB is maximum and, if so, determine an approximate value for this angle. Throughout the following, we note x the length ET , which we seek to determine. The dimensions of the plot are as follows : EM =50 m ; EA =25 m ; AB =5.6 m Note ¸ the measure in radians of the angle ∠ ETA , ˛ the ra-dian measure of the angle ∠ ETB and ‚ the radian measure of the angle ∠ ATB . 1 Using the right-angled triangles ETA and ETB and the lengths provided, express tan ¸ and tan ˛ as a function of x . The tangent function is defined on the interval 0 ; ı 2 by: tan x = sin x cos x 2 Show that the function tan is strictly increasing on the interval 0 ; ı 2 . 3 The angle ∠ ATB admits a measure ‚ belonging to the interval 0 ; ı 2 , an admitted result here that can be ob-served on the figure. We admit that for all real numbers a and b in the interval 0 ; ı 2 : tan a − b = tan a − tan b 1 + tan a × tan b Show that : tan ‚ = 5.6 · x x 2 +765 4 The angle ∠ ATB is maximum when its measure ‚ is max-imum. Show that this corresponds to a maximum on the interval 0 ; 50 of the function f defined by: f ( x ) = 5.6 · x x 2 + 765 Show that there is a single value of x for which the angle ∠ ATB is maximum and determine this value of x to the nearest metre and a measure of the angle ∠ ATB at 0.01 radian. 10. Equations E.5482 In the plane provided with a reference frame O ; I ; J , con-sider the trigonometric circle shown below : 1 a On the trigonometric circle, place the two points M and M hav-ing abscissa − 2 2 . b In the main measurement interval, solve the equa-tion : cos x = − 2 2 2 In the main measurement interval, solve the following equations : a sin x = 1 2 b cos x = 1 2 c sin x = − 3 2 3 Solve in R , the following equation : cos x = 3 2 E.2624 Solve the following equations in R : a sin x = 3 2 b cos x = 2 2 https://chingmath.fr 01234-11ij chapExoCorrec/6823 sacados/6823 ABETMTerrain vu du dessusLimiteduterrainLignemédianex chapExoCorrec/5482 sacados/5482 12√22√3212√22√32-12-√22-√32-12-√22-√32 chapExoCorrec/2624 sacados/2624
OIJC45oNOIJC60oM E.2874 1 Solve in the set − ı ; ı of principal measurements, the following equations : a cos x = 2 2 b sin x = − 1 2 c sin x = 3 2 d cos x = − 1 2 2 Solve the following equations in R : a cos x = 3 2 b sin x = − 2 2 E.3110 Consider the two trigonometric circles below : 1 Give, in the reference frame O ; I ; J , the coordinates of the points M and N . 2 In the interval − 180 o ; 180 o , solve the following equa-tions : a cos x = 1 2 b sin x = √ 2 2 c sin x = − 1 2 3 In the interval − 180 o ; 180 o , solve the following equa-tions : a sin x = 1 2 b cos x = √ 2 2 c cos x = − 2 2 4 What can we say about the set of solutions of each of the previous equations, if we look for the measure of the angles in the set R ? 11. Equations E.7726 1 Consider the equation : ( E ): sin x = 1 2 Justify that each element of the set : ı 6 ; 13 · ı 6 ; 25 · ı 6 ; 37 · ı 6 is a solution of the equation ( E ) . 2 Consider the equation : ( F ): cos 2 · x = 1 2 a Justify that each element of the set − ı 6 ; ı 6 is a solution of the equation ( F ) . b For any relative integer k ( k ∈ Z ) , justify that the num-bers − ı 6 + k · ı and ı 6 + k · ı are solutions of the equation ( F ) . c Deduce the values of the four solutions of equation ( F ) belonging to the interval of principal measurements − ı ; ı . E.7703 Solve the following equations in the range − ı ; ı of principal measurements : a sin x + ı 4 = √ 3 2 b cos 2 x + ı 3 = 2 2 E.7725 In the interval − ı ; ı of the main measurements, solve the following two equations : a cos 2 · x = 1 2 b sin 2 · x = √ 3 2 E.8203 Solve, in the range − ı ; ı , the fol-lowing equations : a sin 2 · x = 2 2 b cos 3 · x + ı 3 = 1 2 https://chingmath.fr chapExoCorrec/2874 sacados/2874 sacados/3110 OIJC45oNOIJC60oM chapExoCorrec/7726 sacados/7726 chapExoCorrec/7703 sacados/7703 chapExoCorrec/7725 sacados/7725 sacados/8203