Grade 11 / Trigonometry 27 exercises (including 26 corrected)

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ABCD60o¸4cm8;9cm ¸˛ACB OIJCM¸ ChingQuizz : 1 exercise available for Quizz assessment : 1. Reminders E.6038 Consider the triangle ABC right-angled B shown below : 1 Determine the length of segment [ BC ] rounded to the nearest millimetre. 2 Deduce the measure of the angle ∠ CDB rounded to the nearest degree. E.2182 Consider a triangle ABC right-angled C . We note : ¸ = CAB ; ˛ = ABC 1 Justify that the angles CAB and CBA are two comple-mentary angles. 2 a Using the lengths of the sides of the triangle ABC , express the values of cos ¸ and sin ˛ . b Deduce the equality: cos ¸ = sin ı 2 − ¸ 3 a Using the lengths of the sides of the triangle ABC , express the values of tan ¸ and tan ˛ b Deduce the equality: tan ı 2 − ¸ = 1 tan ¸ 4 Establish equality: cos ¸ 2 + sin ¸ 2 = 1 2. Radians E.534 In an orthonormal frame of ref-erence ( O ; I ; J ) , consider a circle of center O and radius 1 (this circle passes through the points I and J ) . A point M of the circle is marked by the measure of the angle MOI 1 a Give the measure of the circumference of the circle. b Complete the following table : Valeur de ¸ 0 360 180 90 Arc length IM c What can we say about the table above? 2 Using proportionality, complete the table below : Valeur de ¸ 36 45 60 30 Arc length IM E.2203 1 Determine the exact radian measure of the following an-gles : a 90 o b 60 o c 45 o d 30 o e 72 o f 1 o 2 Determine the exact measure in degrees of the following angles : a ı 2 rad b ı 3 rad c ı 6 rad d 3 · ı 5 rad e ı 12 rad f 3 · ı 4 rad 3 Complete the blanks below with the appropriate values, approximated to the nearest thousandth : a 66 o ≈ : : : rad b 137 o ≈ : : : rad c 2 rad ≈ : : : o d 0.69 rad ≈ : : : o https://chingmath.fr chapExoCorrec/6038 sacados/6038 ABCD60o¸4cm8;9cm chapExoCorrec/2182 sacados/2182 ¸˛ACB chapExoCorrec/534 sacados/534 OIJCM¸ chapExoCorrec/2203 sacados/2203
0ı2ı120oı2 0ı2ı135oı2 OaIABObIABCOcIABCD OdIABCDEOeIABCDEFOfIABCDEFG OgIABCDEFGHOhIABCDEFGHJOiIABCDEFGHJK ABCI1cm E.2721 Below are two graduated lines rep-resenting the measurements of an angle in radians on the interval 0 ; 2 ı . 1 2 Complete the bottom scale (representing an angle measure-ment in radian) , then complete the top values representing the corresponding conversion to degree : E.2188 The first nine regular polygons in-scribed in the trigonometric circle are shown below. 1 Give the measure, in radians, of the angle at the cen-ter separating two consecutive vertices of each of these polygons: 2 Name each of these polygons. 3. Remarkable angles E.2180 Let ABC be an equilateral tri-angle whose side measures 1 cm . Note I the middle of segment [ BC ] . 1 What does the straight line ( AI ) represent in the trian-gle ABC ? 2 Complete the table below : CIA CAB CAI IAC Measurement in radian 3 a Using the Pythagorean theorem, show that : AI = 3 2 cm . b In the triangle AIC , determine the sine, cosine and tangent of the angles IAC and ICA . Then complete the following table : ¸ ı 6 rad ı 3 rad cos ¸ sin ¸ tan ¸ https://chingmath.fr chapExoCorrec/2721 sacados/2721 0ı2ı120oı2 0ı2ı135oı2 chapExoCorrec/2188 sacados/2188 OaIABObIABCOcIABCD OdIABCDEOeIABCDEFOfIABCDEFG OgIABCDEFGHOhIABCDEFGHJOiIABCDEFGHJK chapExoCorrec/2180 sacados/2180 ABCI1cm
ABC1cm ABHIOCDEFGM OIJCMNMxMyNy¸o E.2181 Consider the right-isosceles triangle at C such that BC =1 cm 1 Complete the following table : ACB CAB Mesure en radian 2 a Using the Pythagorean theo-rem, determine the measure of side [ AB ] . c In the right-angled triangle ABC , determine the sine, cosine and tangent of the angle CAB , then complete the following table : ¸ cos ¸ sin ¸ tan ¸ ı 4 rad E.10534 Proposal: trigonometric table of notable angles : ¸ 0 ı 6 ı 4 ı 3 ı 2 cos ¸ 1 √ 3 2 √ 2 2 1 2 0 sin ¸ 0 1 2 √ 2 2 √ 3 2 1 tan ¸ 0 √ 3 3 1 √ 3 × 4. Remarkable angles E.7550 Consider the figure below, où C is a semicircle of center O and admitting the segment [ IA ] for diameter : The other points on this figure belong to the semicircle C and verify the following properties : The triangle OAD is an equilateral triangle; triangle OCM is an isosceles right triangle in C ; The triangle AEO is a right-angled triangle in O ; The half-right [ OB ) is the bisector of the angle ∠ DOA ; The point F is the symmetrical of the point D with re-spect to the line ( EO ) ; The measures of the angles ∠ AOG and ∠ AOC are sup-plementary; The point H is the point of intersection of the semicir-cle C with the line parallel to the line ( AI ) and passing through the point B . Give the exact measure of the angles below in radian : a ∠ AOB b ∠ AOC c ∠ AOD d ∠ AOE e ∠ AOF f ∠ AOG g ∠ AOH h ∠ AOI 5. Introduction to the trigonometric circle E.533 Consider the plane provided with the orthonormal reference frame O ; I ; J . Let C be the circle of center O and of radius 1 : this circle is called the trigonometric circle . https://chingmath.fr chapExoCorrec/2181 sacados/2181 ABC1cm sacados/10534 chapExoCorrec/7550 sacados/7550 ABHIOCDEFGM chapExoCorrec/533 sacados/533 OIJCMNMxMyNy¸o
OIJCMNMxMyNy¸o Consider the tangent (Δ) to the circle C passing through the point I and perpendicular to the x-axis. We place a point M on the circle C , note : This point is marked by the angle ¸ = IOM M x the orthogonal project of M on the ( OI ) axis; M y the orthogonal project of M on the ( OJ ) axis; The point M is thus identified by the angle it defines : note M ( ¸ ) , or by its Cartesian coordinates M ( M x ; M y ) . The point N , if it exists, is the intersection of the line (Δ) with the line ( OM ) . We note : N y the orthogonal project of N onto ( OJ ) ; 1 We place ourselves in the triangle OMM x : a What is the nature of the triangle OMM x ? Justify. b Establish the following equalities: cos ¸ = OM x ; sin ¸ = MM x 2 In the triangle ONI rectangular in I , establish the fol-lowing equality: tan ¸ = NI 3 Relative to the angle ¸ , say what the lengths OM x , OM y and ON y represent. 4 In view of the work done previously, justify the equality: cos ¸ 2 + sin ¸ 2 = 1 E.2183 Consider the plane provided with the orthonormal refer-ence frame O ; I ; J . Let C be the circle of center O and of radius 1 : this circle is called the trigonometric cir-cle . Consider the tangent (Δ) to the circle C passing through the point I and perpendicu-lar to the x-axis. We place a point M on the circle C , note : This point is marked by the angle ¸ = IOM M x the orthogonal project of M on the ( OI ) axis; M y the orthogonal project of M on the ( OJ ) axis; The point M is thus identified by the angle it defines : note M ( ¸ ) , or by its Cartesian coordinates M ( M x ; M y ) . The point N , if it exists, is the intersection of the line (Δ) with the line ( OM ) . We note : N y the orthogonal project of N onto ( OJ ) ; 1 We place ourselves in the triangle OMM x : a What is the nature of the triangle OMM x ? Justify. b Establish the following equalities: cos ¸ = OM x ; sin ¸ = MM x 2 In the triangle ONI rectangular in I , establish the fol-lowing equality: tan ¸ = NI 3 Relative to the angle ¸ , say what the abscissa of the point M , the ordinate of the point M and the ordinate of the point N represent. 4 Establish the identity: cos ¸ 2 + sin ¸ 2 = 1 https://chingmath.fr chapExoCorrec/2183 sacados/2183 OIJCMNMxMyNy¸o
OIJCMNxyz¸oMNxyz180−¸o OIJCMNxyz¸o OIJCM53oM−30o− OIJCM220o−140o E.597 On the trigonometric circle, the point M is marked from the angle ¸ and the point M is marked using the angle ¸ +90 . 1 a Give as a function of ¸ , the measure of the angle ∠ x OM . b Deduce the measure of the angle ∠ M OJ . 2 Find relationships between the values of cos ¸ , cos( ¸ + 90) , sin ¸ and sin( ¸ +90) . E.596 Consider the plane equipped with the orthonormal coordinate system O ; I ; J . Let C be the circle with center O and radius 1: this circle is called the trigono-metric circle . Consider the line (Δ) passing through the point I and perpendicular to the x-axis. Consider point M in the plane that lies on circle C ; angle MOI measures ¸ degrees. 1 Give a relationship involving the length x and the mea-sure of angle ¸ . 2 Do the same with the length y and angle ¸ . 3 a By examining the ratio NI OI , derive a relationship on the unit circle between the length z and the angle ¸ . b What is the name of line (Δ) relative to circle C ? E.595 The plane is provided with an or-thonormal reference frame ( O ; I ; J ) . The circle with center O the origin of the reference frame and radius 1. is called the trigonometric circle To characterize any point M of the circle, we measure the angle MOI . This is the be-ginning of polar markers, where each point will be uniquely characterized relative to their distance from the origin of the marker and the aforementioned angle. An angle will have a positive value if the arc IM is traversed in an anti-clockwise direction. And the angle will be negative, if the same arc is traversed in a clockwise direction. Any point on the circle can be marked by an angle. But two angles can easily be associated with a single point : In the same way, we can also speak of an angle making more than one turn. But, since plane geometry doesn’t study dy-namic figures, but rather fixed configurations of the plane, talking about an angle of 390 o and 30 o amounts to the same thing What can we say about the points characterized by the an-gles : 30 o ; − 330 o ; 390 o https://chingmath.fr chapExoCorrec/597 sacados/597 OIJCMNxyz¸oMNxyz180−¸o chapExoCorrec/596 sacados/596 OIJCMNxyz¸o chapExoCorrec/595 sacados/595 OIJCM53oM−30o− OIJCM220o−140o
OIJπ6π4π32π33π45π6-π6-π4-π3-2π3-3π4-5π6 OIJMπ6Nπ3 6. Associated angles E.7704 The plane is given an orthonormal coordinate system O ; I ; J and the trigonometric circle be-low is considered : où are represented the points M of the trigonometric circle whose principal measure of the oriented angle −→ OI ; −−→ OM is a remarkable angle. Give the exact value of the ratios below : a cos ı 6 b cos ı 4 c cos 5 ı 6 d cos ı e sin − ı 4 f sin 2 ı 3 g sin − 5 ı 6 h sin ı 2 E.2871 1 Draw a trigonometric circle and place the following points, marked by their principal measure : a A 2 ı 3 b B − 3 ı 4 c C 5 ı 6 d D ı 4 e E − ı 4 f F − ı 6 2 Specify the cosine and sine values associated with each of the angles locating the previous points. E.2179 Consider the trigonometric circle C in the plane provided with a reference frame O ; I ; J 1 a Determine the Carte-sian coordinates of the point M . b Place the point M symmetrical to the point M by symmetry of axis ( OJ ) . Give the Cartesian coordinates of the point M . Then, give the angle locating the point M in the circle C . c Locate the point M  symmetrical to the point M by symmetry of axis ( OI ) . Give the Cartesian coordi-nates of the point M  . Then give the angle locating the point M  in the circle C . 2 a Determine the Cartesian coordinates of the point N . b Place the point N symmetrical to the point N by sym-metry of axis ( OJ ) . Give the Cartesian coordinates of the point N . Then give the angle locating the point N in the circle C . c Place the point N  symmetrical to the point N by sym-metry of axis ( OI ) . Give the Cartesian coordinates of the point N  . Then give the angle locating the point N  in the circle C . https://chingmath.fr chapExoCorrec/7704 sacados/7704 OIJπ6π4π32π33π45π6-π6-π4-π3-2π3-3π4-5π6 chapExoCorrec/2871 sacados/2871 chapExoCorrec/2179 sacados/2179 OIJMπ6Nπ3
aOIJM¸−¸MbOIJM¸McOIJM¸MdOIJM¸−¸M aOIJMMbOIJMMcOIJMMdOIJMM aOIJMxyMbOIJMxyMcOIJMxyMdOIJMxyM OIJCMNxyz¸oMNxyz−¸o E.6574 1 In the following four cases, a point M is placed on the trigonometric circle marked by an angle ¸ . Recall that we then note : ∠ IOM = ¸ or M ¸ . From this point M is placed a new point M : Express the angle marking the point M as a function of ¸ . 2 We will use the following definition and properties : Definition: Two triangles are isometric if their sides are two by two of the same measure. Proposition: If two triangles have a side of the same length adjacent to two respectively equal angles then these two triangles are isometric Justify, in each case, that the triangle shown in solid line and the triangle shown in dotted line are isometric. 3 Open the file ˇ angleAssocie.ggb ı. Change the position of point M and observe the relation-ship between the coordinates of point M and M in each case. 4 Indicate on the figure the coordinates of the point M as a function of the coordinates ( x ; y ) of the point M : E.591 On the trigonometric circle, we mark the point M and M relatively from the ∠ IOM and ∠ IOM ’they form from the x-axis. 1 Compare : cos ¸ et cos( − ¸ ) . 2 Compare : sin ¸ et sin( − ¸ ) . 3 Compare : tan ¸ et tan( − ¸ ) . 7. Associated angles 2 https://chingmath.fr chapExoCorrec/6574 sacados/6574 aOIJM¸−¸MbOIJM¸McOIJM¸MdOIJM¸−¸M aOIJMMbOIJMMcOIJMMdOIJMM aOIJMxyMbOIJMxyMcOIJMxyMdOIJMxyM chapExoCorrec/591 sacados/591 OIJCMNxyz¸oMNxyz−¸o
OIJCMNMxMyNy¸MNMxMyNy¸ı2 OIJCM¸M OIJCM¸M OIJCM¸M OIJCM¸M E.2235 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 Simplify each of the following expressions : a cos x − ı b sin x − ı 2 c sin x + ı 2 d cos x + ı 2 E.2229 Simplify the writing of each of the ex-pressions below : a sin 3 ı + x b cos 5 ı 2 − x c cos x − ı 2 d cos ı 2 + x E.8576 Simplify the writing of each of the ex-pressions below : a sin ı − x + cos ı 2 − x b 3 · sin ı + x − 2 · sin ı − x + 4 · sin x − ı E.2244 1 We give the exact value : cos ı 8 = 2+ 2 2 . a Using the formula cos x 2 + sin x 2 =1 , determine the exact value of sin ı 8 . b Deduce the exact value of cos 5 ı 8 , justifying your ap-proach. c Établir l’égalité: tan ı 8 = 3 − 2 2 . 2 Consider the following expression : A = cos 9 ı 8 − 3 · sin 5 ı 8 + 2 · cos 7 ı 8 Determine a writing of the expression of A as a function of the trigonometric ratios of the angle ı 8 . 8. Relationship between cosine and sine E.528 In the plane provided with an orthonormal ref-erence frame and for ¸ ∈ R , consider two points M ( ¸ ) and M ¸ + ı 2 of the trigonometric circle. 1 Using the coordi-nates of the points shown on the figure, give the cosine, sine and tangent values for the angles ¸ and ¸ + ı 2 . 2 a By what transformation, the triangle OMM x has 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 ¸ : cos ¸ sin ¸ tan ¸ ¸ ∈ 0 ; ı 2 ¸ ∈ ı 2 ; ı ¸ ∈ − ı ; − ı 2 ¸ ∈ − ı 2 ; 0 https://chingmath.fr chapExoCorrec/2235 sacados/2235 chapExoCorrec/2229 sacados/2229 chapExoCorrec/8576 sacados/8576 chapExoCorrec/2244 sacados/2244 chapExoCorrec/528 sacados/528 OIJCMNMxMyNy¸MNMxMyNy¸ı2 OIJCM¸M OIJCM¸M OIJCM¸M OIJCM¸M
OIJCMMxMy¸oMMxMy˛oP(d OIJCMMxMyNxNy¸(dN˛ OIJCMMxMyNxNy¸(dN˛ 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 will 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 , express the area of the triangle ONN in another way. 6 Establish the following formula : IN + tan ¸ = 1 cos ¸ × 1 sin ¸ 7 Deduce the relationship : tan ¸ + ı 2 = − 1 tan ¸ . E.529 In the orthonormal refer-ence frame ( O ; I ; J ) , con-sider the circle C trigono-metric: i.e. the circle with center O and radius 1. Note the points M and M respectively marked by the complementary angles ¸ and ˛ ¸ + ˛ = ı 2 . We note : M ( ¸ ) and M ( ˛ ) Note ( d ) 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 symmetric of the point M relative to the line ( d ) : a Justify that the point N is a point on the circle C . b Give the measure of the angle JON as a function of ¸ . Deduce that the point N belongs to the half-line [ OM ) . c Justify that the symmetrical of M relative to the line ( d ) is the point M . Note M x (resp. M x ) the orthogonal project on the x-axis and My (resp. M y ) the orthogonal project on the ordinate axis of the point M (resp. M ) 2 a Establish links between 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 finish 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 on the trigonometric circle. Consider two points M and N on the trigonometric cir-cle, characterized respectively by the angles ¸ and ˛ and their respective orthogonal projections on the axes of the reference frame : https://chingmath.fr chapExoCorrec/529 sacados/529 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 straight 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 linking tan ¸ and tan ı 2 − ¸ . E.538 Consider an orthonormal coor-dinate system O ; I ; J ) and the trigonometric circle of this coordinate system : that is, the circle with center O and ra-dius 1. The 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 angles ¸ and − ¸ : a Draw the orthogonal projection of M onto the line ( OI ) . Label it M x . b Draw the orthogonal projection of M onto the line ( OJ ) . Label it M y . c Label N as the point of intersection of lines ( OM ) and (Δ) . Draw the orthogonal projection of N onto line ( OJ ) . Name it N y . d Do the same for point M (by placing points M x , M y , N , N y ) . 2 a Compare the x-coordinates 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( − ¸ ) ?. https://chingmath.fr OIJCMMxMyNxNy¸(dN˛ chapExoCorrec/538 sacados/538 OIJCM¸oM−¸o