Outside the high school program / Trigonometry 6 exercises (100% corrected)

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ABC30o√12EFG45o√18 1. Remarkable angles E.747 1 a Draw the triangle ABC equilateral of sides 6 cm . b Trace the perpendicular bisector ( d ) of the segment [ AC ] with the compass. Name the point I , the point of intersection of ( d ) with [ AC ] . c Verify that the median is also the height. 2 Give all the exact values of the lengths of the triangle AIB . 3 Give the measure of each of the angles of the triangle AIB . Justify, then mark them on your drawing. 4 Complete the following table with exact values : 30 o 60 o sin cos tan E.869 1 Draw a circle C of 4 cm radius and an equilateral triangle ABC inscribed in the circle C . 2 Using the ungraduated ruler and compass, find the point I midpoint of the segment [ AB ] : leave a trace of your constructions 3 a What is the nature of the triangle AIC ? Justify. b Give the measure of the angle ICA . Justify. 4 a Show that : CI =6 cm . b Posing AB = x , justify that : AI = 1 2 x . Then show that : x = 4 3 c Calculate the exact value of the sine of 30 o . Optional question : 5 Using the formula (cos x ) 2 +(sin x ) 2 =1 , determine the value of the cosine of 30 o . E.3876 Consider the two triangles ABC and EFG rectangles respectively in C and G shown below : 1 a Using the supplementary angles of a triangle, deter- mine the measure of angle ABC . b Determine the lengths of BC and AC . 2 a Justify that triangle EFG is an isosceles right trian-gle. b Determine the length of EG . Here is a table of trigonometric ratios for notable angles : 0 o 30 o 45 o 60 o 90 o sin 0 1 2 √ 2 2 √ 3 2 1 cos 1 √ 3 2 √ 2 2 1 2 0 tan 0 √ 3 3 1 √ 3 E.745 Let ABC be an isosceles right-angled triangle at B and such that AB =4 cm . 1 Draw the figure, then give, without justification, the value of each of the angles of the ABC triangle. 2 Give the exact value of the tangent of an angle of 45 o . 3 a Give the exact measure of segment [ AC ] . b Give the exact value of the sine and cosine of 45 o . E.3908 The unit of length is the centime-ter. 1 Construct a triangle DOS such that : DS = DO = 6 cm ; ODS = 120 o What is the nature of triangle DOS ? Justify your an-swer. 2 In triangle DOS , draw the height from D . It intersects [ OS ] at H . a Place point H on the figure. b Calculate the exact value of OH . c Deduce that : OS =6 3 cm . 3 In triangle DOS , the height from S intersects line ( OD ) at point I . a Place point I on the figure. b Determine the exact value of OI . The table below gives the exact values of the trigonometric ratios for the notable angles : x sin x cos x tan x 30 o 1 2 3 2 3 3 45 o 2 2 2 2 1 60 o 3 2 1 2 3 https://chingmath.fr chapExoCorrec/747 sacados/747 chapExoCorrec/869 sacados/869 chapExoCorrec/3876 sacados/3876 ABC30o√12EFG45o√18 chapExoCorrec/745 sacados/745 chapExoCorrec/3908 sacados/3908 Groupe Ouest septembre 2003
ABCIx 2. Unclassified financial years E.530 Let ABC be an equilateral tri-angle whose side measures x . Let I be the midpoint of seg-ment [ BC ] . 1 What does the straight line ( AI ) represent in the trian-gle ABC ? 2 Complete the table below : CIA CAB CAI ICA Measurement in degrees 3 a Give the measure of segment [ CI ] as a function of x . b Using Pythagoras’ theorem, determine the measure of segment [ AI ] as a function of x . c In the triangle AIC , determine the sine, cosine and tangent of the angles IAC and ICA . Then complete the following table : ¸ cos ¸ sin ¸ tan ¸ 60 o 30 o https://chingmath.fr chapExoCorrec/530 sacados/530 ABCIx