Outside the high school program / Oriented angles 46 exercises (including 44 corrected)

a
OIJM1M2M3M4 OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ 1. Interval of oriented angles E.2199 Throughout this exercise, the trigono-metric circle has been divided into 12 equal parts. 1 Determine the measure, in radians, of the angles −→ OI ; −−→ OM i for i =1 ;:::; 4 . 2 For each question, a part of the trigonometric circle has been highlighted. Write, as an interval or a union of in-tervals, the set of measures of the angle −→ OI ; −−→ OM when M describes each of these parts : a b c d e f 3 For each question, highlight the set of points M of the circle whose angle −→ OI ; −−→ OM belongs to the indicated interval: a b ı 6 ; 5 ı 6 − ı 2 ; − ı 6 c d 5 ı 6 ; 7 ı 6 0 ; ı 2 ⋃ 5 ı 6 ; ı e f 5 ı 6 ; ı ⋃ − ı ; − 5 ı 6 5 ı 2 ; 29 ı 6 E.2307 1 a For each question, represent on the circle the set of points marked by an angle belonging to the given in-terval: I = 11 ı 6 ; 13 ı 6 J = 2 ı 3 ; 7 ı 3 b Give the expression for each of its intervals using in-terval meetings expressed by principal measures. 2 Associate each of the intervals in the top line with a set in the bottom line: a 2 ı 3 ; 7 ı 6 b 23 ı 6 ; 14 ı 3 b ig ] c − 4 ı 3 ; ı 6 ¸ − ı ; ı 6 ] ∪ 2 ı 3 ; ı ˛ − ı ; − 5 ı 6 ] ∪ 2 ı 3 ; ı ‚ − ı 6 ; 2 ı 3 E.2816 Consider the two real number intervals : a I = − 19 ı 6 ; − 11 ı 4 b J = 20 ı 3 ; 49 ı 6 Assume that these two intervals represent oriented angle mea-sures ; express each of these two intervals using the associated principal angle measures. https://chingmath.fr chapExoCorrec/2199 sacados/2199 OIJM1M2M3M4 OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ OIJ chapExoCorrec/2307 sacados/2307 OIJ OIJ chapExoCorrec/2816 sacados/2816
ABMNP AB 2. Geometric locus and oriented angles E.2213 Consider, in the plane, five points M , A , P , B , N aligned in this order. 1 Determine the measure of the following angles : a MAMB b PAPB c NANB d ABMP e ABPM 2 On the figure above, determine the geometric locus of the Q verifying respectively: a −→ QA ; −−→ QB = ı + 2 kı b −→ QA ; −−→ QB = 0 + 2 kı E.2202 Let A and B be two fixed points in the plane. Determine the geometric locus of points M verifying the following relationships : make a representation of such a situation, specifying the possible locations of point M . a −−→ MA ; −−→ MB = ı + 2 kı b −−→ MA ; −−→ MB = 0 + 2 kı c −−→ MA ; −−→ MB = − ı 2 +2 kı d −−→ MA ; −−→ MB = − ı 2 + kı e −−→ MA ; −−→ MB = ı + kı E.2239 Two points A and B of the plane are considered below : All requested constructions must be carried out using the com-pass and the unengraved ruler only. Construction lines must remain on the figure. 1 Place a point C verifying: −→ CA ; −−→ CB = ı 3 2 Draw the circumscribed circle of the triangle ABC . 3 Highlight the locus of points M verifying the following relationship : −−→ MA ; −−→ MB = ı 3 Which theorem is used? https://chingmath.fr chapExoCorrec/2213 sacados/2213 ABMNP chapExoCorrec/2202 sacados/2202 chapExoCorrec/2239 sacados/2239 AB
OIJ ABC ABCDEI E.2212 1 a Complete the table below : k − 2 − 1 0 1 2 ı 3 + kı b On one of the trigonometric circles below, represent the set of points M verifying the relationship : −→ OI ; −−→ OM = ı 3 + kı where k ∈ Z . 2 In each case, represent the set of points M verifying the relationship specified : a −→ OI ; −−→ OM = ı 2 + kı 3 b 2 −→ OI ; −−→ OM = − 2 ı 3 + kı The following circles have been divided into twelve equal parts. E.2243 Consider the following expressions, where k is a relative integer: a ı 2 + kı b ı 4 + kı 2 c ı + kı 3 d 3 ı 4 + 2 kı 3 In each case, determine the set of points M , when k describes Z , of the trigonometric circle marked by this measure of angle . E.2211 A circle of diameter [ AB ] is shown below. 1 Let M be a point on the semicircle highlighted in bold on the figure. Give the measure of −−→ MA ; −−→ MB . 2 Determine the set of points N of the plane verifying: −−→ NA ; −−→ NB = ı 2 + 2 kı . 3 Which two theorems are useful for this exercise? 3. Plane geometry and Chasles relationship E.2220 cm considers the configuration below where ABC is an equilateral triangle, DCA and CEB are right-angled isosceles triangles at D and E respectively. 1 Give the principal measure in radians of the following angles : a −−→ AD ; −→ AC b −→ AC ; −−→ AB c −−→ BA ; −−→ BE d −−→ DC ; −−→ AD 2 a Justify that ( DB ) is the height of the ADC triangle from D and of the ABC triangle from B . b Give the principal measure in radians of the following angles : −→ IA ; −→ IB ; −→ IB ; −→ IE 3 Using the Chasles relation, determine the principal mea-sure in radians of the following angles : a −−→ CD ; −−→ AB b −−→ DC ; −−→ AB c −→ IB ; −−→ CE https://chingmath.fr chapExoCorrec/2212 sacados/2212 OIJ OIJ OIJ chapExoCorrec/2243 sacados/2243 chapExoCorrec/2211 sacados/2211 ABC chapExoCorrec/2220 sacados/2220 ABCDEI
ABCDEFJI OIJ E.2797 Consider an equilateral triangle AEC inscribed in the rectangle AEFD . Inside the rectangle, we draw the equilateral triangle DAJ ; we note I are center. Please note that the figure below has not been drawn cor-rectly; the aim of the exercise is to show that the points I and J belong to the segments [ AC ] and [ BC ] respectively. We’ll use the following property: −→ u ; −→ v = −→ u ; −→ w = ⇒ −→ v and −→ w are collinear and in the same direc-tion. 1 a Justify that : −→ AC ; −−→ AD = ı 6 . b Justify that the angle at the center −→ IA ; −→ ID mea-sures 2 ı 3 . c Deduce that the vectors −→ AI and −→ AC are collinear. 2 a Justify that the triangle DCJ is isosceles at C . b Deduce the measure of the angle −→ CA ; −→ CJ . c Deduce that the points J , E , C are aligned. 4. Equations and congruences E.2226 1 Solve in the interval − ı ; ı of the main measurements the following equations : a 2 sin 2 x = 1 b cos 3 x = 1 2 Solve in the interval − ı ; ı of principal measurements the following equations : a sin 2 x = sin x b cos 2 x = cos x E.2184 In this exercise, we wish to show that all numbers in the set E = ı 3 + 2 kı 3 ⏐ ⏐ ⏐ k ∈ Z vérifient l’équation: sin 2 x = sin x 1 a Complete, in the table below, the value of ¸ when k runs through the proposed values : k -2 -1 0 1 2 3 4 ¸ = ı 3 + 2 kı 3 b Using the previous question, give the principal mea-sure of the angle ¸ as a function of k : k -2 -1 0 1 2 3 4 Mesure principale de ¸ c Place all points of E on the trigonometric circle below : 2 Show that all elements of E ver-ify the equation : sin 2 x = sin x E.2270 Consider the set E of numbers defined by: E = ı 6 + k · ı 2 ⏐ ⏐ k ∈ Z 1 Give the set of numbers formed by the set of principal measures of the angles in the set E . 2 Show that all the numbers in E verify the following equa-tion : cos(3 x ) = cos x − 2 ı 3 E.2625 When k describes the set Z , then the expression ı 4 + k · ı 2 describes a set of numbers which we note E and which can be written as : E = ı 4 + k · ı 2 ⏐ ⏐ ⏐ k ∈ Z 1 Give the principal measures of the angles represented by this set. 2 Verify that each number in the set E verifies the equa-tion : cos 2 x =0 E.2626 Solve the following equations in the range − ı ; ı of principal measurements : a 2 · cos 2 x = 1 b sin 3 x = 3 2 c cos 2 x = cos x + ı 3 d sin 3 x = cos x E.2967 1 a Solve equation : 2 x 2 + 7 x + 3 = 0 . b Solve the equation below in − ı ; ı : sin 2 x = − 1 2 2 Deduce the set of solutions to the equation : 2 · sin 2 x 2 + 7 · sin 2 x + 3 = 0 https://chingmath.fr chapExoCorrec/2797 sacados/2797 ABCDEFJI chapExoCorrec/2226 sacados/2226 chapExoCorrec/2184 sacados/2184 OIJ chapExoCorrec/2270 sacados/2270 chapExoCorrec/2625 sacados/2625 chapExoCorrec/2626 sacados/2626 chapExoCorrec/2967 sacados/2967
OIJ12√22√32-12-√22-√3212√22√32-12-√22-√32 OIJCMNMxMyNy¸MNMxMyNy¸ı2 OIJCM¸M OIJCM¸M OIJCM¸M OIJCM¸M 5. Equations E.2227 With the help of a graphical representa-tion of the trigonometric circle, determine the set of solutions in the interval − ı ; ı of the main measurements the follow-ing inequalities: a cos x √ 2 2 b 2 sin x 1 c cos x < − 1 2 E.2231 Two trigonometric circles are shown be-low, with remarkable values indicated on the axes : a b 1 On the x-axis, plot the sets of numbers defined by the following interval images : a cos ı 4 ; 5 ı 6 b cos − ı 4 ; 2 ı 3 2 On the x-axis, plot the sets of numbers defined by the following interval images : a sin ı 4 ; 5 ı 6 b sin − ı 4 ; 2 ı 3 E.2232 With the help of a graphical representa-tion of the trigonometric circle, determine the set of solutions in the interval − ı ; ı of the main measurements the follow-ing inequalities: a cos x > 0 b sin x > 0 c sin x < − 1 2 E.2303 1 Using the representation of the trigonometric circle: a Represent on the x-axis the set E : E = cos x ⏐ ⏐ x ∈ ı 6 ; 2 ı 3 Express the set E as an interval. b Represent on the ordinate axis the set F : sin x ⏐ ⏐ x ∈ 2 ı 3 ; 7 ı 3 Express the set F as an interval. 2 Graphically and with the help of the representation of the trigonometric circle, give the set of solutions for each of the inequalities below : a sin x − 2 2 b cos x − 1 2 c cos x < 0 6. Demonstrating the properties of associated angles E.2185 In the plane provided with an orthonor-mal reference frame and for ¸ ∈ R , consider two points M ( ¸ ) and M ¸ + ı 2 of the trigonometric circle. 1 Using the coordinates of the points shown on the figure, give the values of cosine, sine and tangent for the angles ¸ and ¸ + ı 2 . 2 a By what transformation does the triangle OMM x have as its image the triangle OM M y ? b Deduce the values of cos ¸ + ı 2 and sin ¸ + ı 2 as a function of cos( ¸ ) and sin( ¸ ) . 3 We’re going to determine the sign of the different values of the trigonometric functions for the angles ¸ and ¸ + ı 2 . This is to ensure the validity of the formulae found in question 2 whatever the value of the angle ¸ : https://chingmath.fr chapExoCorrec/2227 sacados/2227 chapExoCorrec/2231 sacados/2231 OIJ12√22√32-12-√22-√3212√22√32-12-√22-√32 OIJ12√22√32-12-√22-√3212√22√32-12-√22-√32 chapExoCorrec/2232 sacados/2232 chapExoCorrec/2303 sacados/2303 chapExoCorrec/2185 sacados/2185 OIJCMNMxMyNy¸MNMxMyNy¸ı2 OIJCM¸M OIJCM¸M OIJCM¸M OIJCM¸M
OIJCMMxMy¸oMMxMy˛oP(d OIJCMMxMyNxNy¸(dN˛ OIJCMMxMyNxNy¸(dN˛ cos ¸ sin ¸ tan ¸ ¸ ∈ 0 ; ı 2 ¸ ∈ ı 2 ; ı ¸ ∈ − ı ; − ı 2 ¸ ∈ − ı 2 ; 0 cos ¸ + ı 2 sin ¸ + ı 2 tan ¸ + ı 2 ¸ ∈ 0 ; ı 2 ¸ ∈ ı 2 ; ı ¸ ∈ − ı ; − ı 2 ¸ ∈ − ı 2 ; 0 We deduce that : tan ¸ + ı 2 = sin ¸ + ı 2 cos ¸ + ı 2 = cos ¸ − sin ¸ = − tan ¸ . Subsidiary question: We’ll show in another way that : tan ¸ + ı 2 = − 1 tan ¸ : 4 a Express ON as a function of ¸ . b Using the fact that cos ı 2 − ¸ =sin ¸ , express the value of ON as a function of ¸ c Deduce the area of triangle ONN . 5 Using the fact that ( OI ) is the height of the triangle ONN from O , calculate in a second way the area of the triangle ONN . 6 Establish the following formula : IN + tan ¸ = 1 cos ¸ × 1 sin ¸ 7 Deduce the relationship : tan ¸ + ı 2 = − 1 tan ¸ . E.2186 In the orthonormal reference frame ( O ; I ; J ) , consider the C trigonometric circle: that is, the cir-cle with center O and radius 1. Note the points M and M respectively marked by the com-plementary angles ¸ and ˛ ( ¸ + ˛ = ı 2 ) . Note: M ( ¸ ) and M ( ˛ ) . Let ( d ) be the bisector of the angle JOI and P the point of intersection of C with ( d ) 1 In this question, we will show that the two points M and M are symmetrical relative to the line ( d ) . To do this, let’s denote N the image of point M by the symmetry of axis ( d ) : a Justify that the point N is a point of the circle C . b Give the measure of the angle widehatJON as a func-tion of ¸ . Deduce that the point N belongs to the half-line [ OM ) . c Justify that the image of point M , by symmetry of axis ( d ) , is point M . Let M x (resp. M x ) be the orthogonal project on the x-axis and M y (resp. M y ) the orthogonal project on the y-axis of the point M (resp. M ) . 2 a Relate the coordinates of points M and M in the reference frame ( O ; I ; J ) . b Deduce the following relationships : cos¸ = sin ı 2 − ¸ ; sin ¸ = cos ı 2 − ¸ To complete the study of comparing the cosine and sine of two complementary angles, we also need to see what happens if the point M lies on another quadrant of the trigonometric cir-cle. Consider two points M and N on the trigonometric circle, respectively characterized by angles ¸ and ˛ , and their respec-tive orthogonal projects on the axes of the reference frame : https://chingmath.fr chapExoCorrec/2186 sacados/2186 OIJCMMxMy¸oMMxMy˛oP(d OIJCMMxMyNxNy¸(dN˛ OIJCMMxMyNxNy¸(dN˛
OIJCMMxMyNxNy¸(dN˛ OIJCM¸oM−¸o 3 a For each of the three figures below, orally establish the truth of the assertion below : The angles ¸ and ˛ are complementary if, and only if, the points M and N are symmetrical relative to the line ( d ) b Justify that this is enough for us to establish, for any value of ¸ , the following equalities: cos¸ = sin ı 2 − ¸ ; sin ¸ = cos ı 2 − ¸ . 4 Deduce the relationship between tan ¸ and tan ı 2 − ¸ . E.2187 Consider an orthonormal frame of ref-erence O ; I ; J ) and the trigonometric circle of this frame of reference : i.e. the circle with center O and radius 1. The straight line (Δ) is the tangent at I to the circle C . Let ¸ be any real number. Consider the points M ( ¸ ) and M ( − ¸ ) . An example of this situation is given in the graph opposite. 1 To highlight, the different values of the trigonometric functions associated with the angles ¸ and − ¸ : a Draw the orthogonal project of M on the line ( OI ) . Name it M x . b Draw the orthogonal project of M on the line ( OJ ) . Name it M y . c Name N the point of intersection of the straight lines ( OM ) and (Δ) . Draw the orthogonal project of N onto the line ( OJ ) . Name it N y . d Do the same for point M 2 a Compare the abscissas of points M and M . b Deduce a relationship between cos ¸ and cos( − ¸ ) ?. 3 a Compare the ordinates of points M and M . b Deduce a relationship between sin ¸ and sin( − ¸ ) ?. 4 a Compare the ordinates of points N and N . b Deduce a relationship between tan ¸ and tan( − ¸ ) ?. 7. Polar and Cartesian orientation E.2265 Consider the plane provided with a ref-erence frame ( O ; I ; J ) orthonormal. Each point below is shown with its Cartesian coordinates ( x ; y ) . Determine the polar coordinates [ ;„ ] associated (give a value approximated to the tenth if necessary) : a M − 3 ; − 3 b N (2 ; − 2) c P 6 ; − 2 d Q (5 ; 2) E.2266 Consider the plane provided with a ( O ; I ; J ) orthonormal coordinate system. 1 Each point below is shown with its polar coordinates [ ; „ ] . Determine the Cartesian coordinates of each of its points : (give a value approximated to the tenth if necessary) : a M 3 ; ı 4 b N 2 ; − ı 6 c P 3 2 ; 5 ı 6 d Q 10 ; − ı 12 2 Consider the point R with polar coordinate [4 ; „ ] such https://chingmath.fr OIJCMMxMyNxNy¸(dN˛ chapExoCorrec/2187 sacados/2187 OIJCM¸oM−¸o chapExoCorrec/2265 sacados/2265 chapExoCorrec/2266 sacados/2266
OIJCMı3M−ı3− OIJKLπ6A1A2A3A4A5A6A7A8A9A10A11A12 that tan „ = 3 . Can we uniquely determine the Carte-sian coordinates of the point R . E.2267 In an ( O ; I ; J ) orthonormal coordinate system, construct the following points using only compass and a straightedge : a A 3 ; 3 ı 2 b B 5 2 ; ı 4 c C 2 ; − 2 ı 3 d D 3 ; ı 6 E.2305 1 Consider two points M and N of the plane respectively with polar coordinate 3 ; 4 ı 3 et 2 ; − ı 4 . Determine the Cartesian coordinates of the points M and N . 2 Consider two points P and Q of the plane with Cartesian coordinates ( − 5 ; 5) and 3 ; − 3 respectively Determine the polar coordinates of points P and Q . E.2908 In the plane provided with a reference frame O ; I ; J : 1 Consider the two points A and B defined by their Carte-sian coordinates : a A − 3 3 ; 3 b B − 1 10 ; − 3 10 Determine the polar coordinates of these two points. 2 Consider the two points C and D defined by their polar coordinates : a C 1 4 ; − 3 ı 4 b D 15 ; ı 6 Determine the Cartesian coordinates of these two points. 8. Oriented angles E.810 In the plane provided with an orthonormal reference frame ( O ; I ; J ) , consider the circle with center O and radius 1 called circle trigonométrique . Any point M defines a geo-metric angle IOM . The direction of travel of the trigonometric circle al-lows any point on the circle to be characterized by its geometric angle: the angle is positive if the arc IM is oriented anticlock-wise. the angle is negative if the arc IM is oriented clockwise. In the above representation : On a: −→ OI ; −−→ OM =+ ı 3 rad In the trigonometric circle, we note M + ı 3 . On a: −→ OI ; −−−→ OM = − ı 3 rad In the trigonometric circle, we note M − ı 3 . 1 In the figure below, the points A i define an angle oriented −→ OI ; −−→ OA i having a measure ˇ remar-quable ı. For each of the points, indicate the measure of the associated angle and add the sign to iden-tify each marked point of the trigonometric circle: 2 In the trigonometric circle above, place on this figure the points N , P , Q , R , S , T performing the following measurements : a −→ OI ; −−→ ON = − ı 4 rad b −→ OI ; −−→ OP = 5 ı 6 rad c −→ OI ; −−→ OQ = − 2 ı 3 rad d −−→ OK ; −−→ OR = − ı 4 rad e −−→ OK ; −→ OS = ı 6 rad f −→ OJ ; −→ OT = − ı 4 rad https://chingmath.fr sacados/2267 chapExoCorrec/2305 sacados/2305 chapExoCorrec/2908 sacados/2908 chapExoCorrec/810 sacados/810 OIJCMı3M−ı3− OIJKLπ6A1A2A3A4A5A6A7A8A9A10A11A12
IIJJMNPMNPMNPMNP ABCDI OIAJ −2ı−ı0ı2ı3ı4ıOIJABC E.5464 In the plane provided with a reference frame O ; I ; J , consider the trigonometric circle shown below on which several points are placed : The points M , N , P ver-ify the following measure-ments : IOM = 30 o ; ION = 45 o IOP = 60 o 1 Give the measure of the angles marking the points M , N , P en radians. 2 The points M , N , P sont respectively the images of the points M , N , P par the symmetry of axis ( OI ) : a What can we say about −→ OI ; −−→ OM et −→ OI ; −−−→ OM ? b Give the measure in radians of the following angles : −→ OI ; −−−→ OM ; −→ OI ; −−→ ON ; −→ OI ; −−→ OP 3 Points M  , N  and P  are respectively the images of points M , N , P by the axis symmetry ( OJ ) : a What can be said about −→ OI ; −−→ OM and −→ OI ; −−−→ OM  ? b Give the measure in radians of the following angles : −→ OI ; −−−→ OM  ; −→ OI ; −−−→ ON  ; −→ OI ; −−→ OP  4 Points M  , N  and P  are respectively the images of points M , N , P by the symmetry center O : a What algebraic relationship verifies the two angles : −→ OI ; −−→ OM ; −→ OI ; −−−→ OM  b Give the measure in radians of the following angles : −→ OI ; −−−→ OM  ; −→ OI ; −−−→ ON  ; \quad −→ OI ; −−→ OP  E.5465 Consider the quadrilat-eral ABCD représenté below, which consists of two triangles ABC et ACD respectivement equilateral and isosceles right-angled at D . Using the points on this figure and for each question, give an oriented angle achieving the following measurements : a ı 3 rad b − ı 4 rad c − ı 6 rad d 7 ı 12 rad E.2153 Consider the trigonometric circle C below inscribed with a dodecagon (12-sided regular polygon) 1 Determine the measure of the angle −→ OI ; −→ OA 2 Place on the circle C the points M , N , P such that : a −→ OI ; −−→ OM = 2 ı 3 rad b −→ OJ ; −−→ ON = − ı 6 rad c −→ OA ; −−→ OP = − ı 2 rad c −−→ OQ ; −→ OJ = − 5 ı 6 rad 9. Main measures E.2189 Consider the following grad-uated line on which the trigonometric circle (circle of radius 1) . is placed The point I (unit of abscissas) is placed on the origin of the graduated line. We roll the circle along the entire graduated line. 1 Indicate the various possible positions of the point I on the right. 2 Justify that the points A , B , C of the graduated line represent the same point on the trigonometric circle. https://chingmath.fr chapExoCorrec/5464 sacados/5464 IIJJMNPMNPMNPMNP chapExoCorrec/5465 sacados/5465 ABCDI chapExoCorrec/2153 sacados/2153 OIAJ chapExoCorrec/2189 sacados/2189 −2ı−ı0ı2ı3ı4ıOIJABC
OIJCM76ı−56ı -5ı-4ı-3ı-2ı-ı0ı2ı3ı4ı5ı6ı7ı8ıABC OIJKLπ6π4π32π33π45π6-π6-π4-π3-2π3-3π4-5π6 E.2200 1 a What can you say about the points on the trigono-metric circle marked by the angles : 50 o ; 410 o ; 3650 o ; − 310 o b Same question for angles : ı 3 rad ; − 5 ı 3 rad ; 7 ı 3 rad Thus, for ¸ ∈ R , all angles of the set ¸ +2 k · ı ⏐ ⏐ k ∈ Z define the same point on the trigonometric circle. The figure opposite shows that the point M can be located by the two angles : M 7 6 ı ; M − 5 6 ı The interval − ı ; ı is called the interval of principal mea-sures and has a length of 2 ı . We admit that, for any ¸ · R , there exists a single element of the set ¸ +2 kı | k ∈ Z be-longing to the interval − ı ; ı . 2 a Complete with yes or no the table below : Angle − 3 7 ı − 5 3 ı 7 8 ı 57 4 ı − ı appartient à − ı ; ı b Which of the numbers below belongs to the interval ] − ı ; ı ] ? 9 ı 2 − 4 ı ; 9 ı 2 − 2 ı ; 9 ı 2 ; 9 ı 2 +2 ı ; 9 ı 2 +4 ı c Which of the numbers below belongs to the interval ] − ı ; ı ] ? − 5 ı 3 − 2 ı ; − 5 ı 3 ; − 5 ı 3 +2 ı ; − 5 ı 3 +4 ı d Soit E = 5 ı 4 +2 kı ⏐ ⏐ ⏐ k ∈ Z . Give the unique element of the set E belonging to the interval of principal mea-sures. E.2738 Consider the graduated line below where the points A 20 3 ı are placed, B − 17 5 ı et C 43 8 ı . 1 a Graphically, determine the number of times by which 2 · ı must be removed from the abscissa of point A in order to obtain the primary measure of this num-ber? b Deduce the principal measure of 20 3 . 2 Determine the principal measure of the abscissas of points B and C . E.535 The plane is given an orthonor-mal reference frame O ; I ; J . We denote by M and N two points of the trigonometric circle. 1 Which of the following angle measures belong to the in-terval of principal measures : a 5 ı 3 b − 7 ı 4 c − 2 ı 3 d 1.1 ı 2 Determine the principal measure of the angles defined by the points M , N , P and Q below, then place each of these points on the trigonometric circle opposite : a −→ OI ; −−→ OM = 7 ı 3 b −→ OI ; −−→ ON = − 15 ı 4 c −→ OI ; −−→ OP = 5 ı 3 d −→ OI ; −−→ OQ = 19 ı 6 E.2201 Determine the principal measure of the following oriented angles : a 9 ı 4 b 192 ı 6 c − 5 ı 4 d − 33 ı 2 e 16 ı 7 f 52 ı 3 E.2242 1 Give the principal measure of the following angles : a 15 ı 7 b 13 ı 9 c 173 ı 12 d 165 ı 7 e − 64 ı 15 f − 429 ı 33 2 Here are two intervals of oriented angle measurements : I = 2 ı 3 ; 7 ı 3 ; J = 7 ı 3 ; 35 ı 8 Determine the writing of each of these sets using the principal measures of oriented angles. E.2737 1 In this question, we propose to determine the principal measure of the angle ¸ = 73 5 ı : a Let k be a relative integer achieving the following fram-ing : − ı < 73 5 ı +2 · k · ı ı Make a frame of k using the frame above. b Using the calculator, determine the single integer k achieving this framing. c Deduce the principal measure of the angle ¸ . 2 In the same way, determine the principal measure of the following angles : a − 29 3 ı b − 27 4 ı c 70 9 ı https://chingmath.fr chapExoCorrec/2200 sacados/2200 OIJCM76ı−56ı chapExoCorrec/2738 sacados/2738 -5ı-4ı-3ı-2ı-ı0ı2ı3ı4ı5ı6ı7ı8ıABC chapExoCorrec/535 sacados/535 OIJKLπ6π4π32π33π45π6-π6-π4-π3-2π3-3π4-5π6 chapExoCorrec/2201 sacados/2201 chapExoCorrec/2242 sacados/2242 chapExoCorrec/2737 sacados/2737
OIJ OIJ OIJ IOıaπ2b-π2c-ıdπ3e3π4f-π3g-π4h-5π6j0o20o40o60o80o100o120o140o160o180o160o140o120o100o80o60o40o20o ABCDE5ı75ı9 OBCDEFGAI E.2799 1 Give, in the form of interval meetings, the set formed by the principal measures of the angles marking the high-lighted points of the trigonometric circle: a b 2 For each question, highlight the set of points whose ori-ented angle is the set specified under the trigonometric circle: a 13 ı 3 ; 29 ı 6 b ı 2 ; 8 ı 3 E.2269 Determine the principal measure of the following oriented angles : a 254 ı 5 b − 70 ı 3 c 92 ı 7 E.4920 Consider a graduated line with origin O on which is placed points defined by their abscissa : a ı 2 ; b ı ; c − ı 2 ; d − ı ; e ı 3 f 3 ı 4 ; g − ı 3 ; h − ı 4 ; j − 5 ı 6 Consider the circle C of radius 1 placed on the graduated line as shown in the previous figure. 1 a Soit M un C tel point that the arc OM mesure ı . Give the measure of the angle OIM b Place the single point A of the circle C such that the arc OA ait has length ı . 2 a So M a point of C tel that the arc OM mesure ı 2 . Give the measure of the angle OIM b Place the two points B and C belonging to the circle C tel that the arcs OB et OC aient for length ı 2 . 3 Similarly, place the points E , F , G , H , J tels as the arcs OE , OF , OG , OH , OJ aient respectively the same length as the abscissa of the points e , f , g , h , j . 10. Golden angles and algebra E.2798 The drawing below shows a broken line ABCDE whose geometric angles ABC and BCD have been indicated. The straight lines ( AB ) and ( DE ) are parallel. Determine the principal measure of the angle oriented −−→ DC ; −−→ DE . E.2268 Consider the regular heptagon ABCDEFG of center O . The point I is the point of intersection of the straight lines ( BC ) and ( FG ) . 1 Give, justifying your approach, the measure of the follow-ing oriented angles (we’ll go through the geometric angle first) : a −−→ OG ; −−→ OB b −→ AG ; −−→ AB c −−→ CE ; −−→ CD 2 Determine, with the help of the Chasles relationship, the measure of the following oriented angles : a −−→ OE ; −−→ CB b −→ IG ; −→ IB https://chingmath.fr chapExoCorrec/2799 sacados/2799 OIJ OIJ OIJ OIJ chapExoCorrec/2269 sacados/2269 sacados/4920 IOıaπ2b-π2c-ıdπ3e3π4f-π3g-π4h-5π6j0o20o40o60o80o100o120o140o160o180o160o140o120o100o80o60o40o20o chapExoCorrec/2798 sacados/2798 ABCDE5ı75ı9 chapExoCorrec/2268 sacados/2268 OBCDEFGAI
ABCDEF E.2233 Consider the square ABCD . Let be the point E outside the square such that BCE is equi-lateral. Let F be the point inside the square such that the triangle ABF is equilateral. We wish to show that the points D , F and E are aligned. 1 a Give the measure of the following two oriented an-gles : −→ AF ; −−→ AD ; −−→ DF ; −−→ DA b Deduce the measure of the oriented angle −−→ DC ; −−→ DF . 2 a Give the measure of the angle oriented −−→ CD ; −−→ CE . b Deduce the measure of the angle oriented −−→ DC ; −−→ DE . 3 Deduce that the points D , F and E are aligned. The aim of the following questions is to use the Chasles rela-tion. 4 Determine the measure of oriented angles : a −−→ BE ; −−→ CF b −→ AF ; −−→ CE https://chingmath.fr chapExoCorrec/2233 sacados/2233 ABCDEF