Trigonometry Basics.pdf - AKSHATHA

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TRIGONOMETRY The word trigonometry is originated from the Greek words “tri” means three, “gonia” means angle and “metron” means measure. Hence, the word trigonometry means three angle measure i.e. it is the study of geometrical figures, which have three angles i.e triangles. Systems of measurement of angle 1 1. Degree measure : When the initial ray is rotated through ( )th of one revolution, we say that an 360 angle of one degree (1°) is formed at the initial point. 1 right angle = 90° 1°= 601 (minutes) and 11 = 6011(seconds) Note : This system is also called the sexagesimal system (or) British system 2. Radian Measure : A radian is the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle. It is the written as 1c. Note : *If an arc of length l subtends an angle 𝜃 radians at the centre of circle of radius r, 𝑙 then 𝜃 = 𝑟. This system is also called circular system. 1 * The area of the sector of a circle having a central angle 𝜃 radians is 2 𝑟 2 𝜃. Relation between Degree and Radian 𝜋 Radian measure = × Degree measure 180 180 Degree measure = × Radian measure 𝜋 In particular, 2𝜋 radian=360° , 𝜋 radian=180° Remember : 𝜋 • 1°=180 radians = 0.0175 radians (approximately) 180 • 1c = radians = 57° 171 4411 (approximately) 𝜋 Trigonometric Functions Definitions of trigonometric functions 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑦 sin𝜃 = =𝑟 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 𝑥 cos𝜃 = =𝑟 𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑦 tan𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒 = 𝑥 1 1 1 Note : cosec𝜃 = 𝑠𝑖𝑛𝜃; sec𝜃 = 𝑐𝑜𝑠𝜃 ; cot𝜃 = 𝑡𝑎𝑛𝜃 Signs of the trigonometric functions • In the I-quadrant, all the trigonometric functions are +ve. • In the II-quadrant, sin𝜃 and cosec𝜃 are +ve and the remaining trigonometric functions are –ve. • In the III quadrant, tan𝜃 and cot𝜃 are +ve and the remaining trigonometric functions are –ve. • In the IV-quadrant, cos𝜃 and sec𝜃 are +ve and the remaining trigonometric functions are –ve. Trigonometric functions of standard angles 0° 30° 45° 60° 90° 180° 270° 360° 1⁄ 1⁄ √3⁄ Sin𝜃 0 2 1 0 -1 0 √2 2 √3⁄ 1⁄ 1⁄ Cos𝜃 1 2 0 -1 0 1 2 √2 1⁄ Not Not Tan𝜃 0 1 √3 0 0 √3 defined defined Not 2⁄ Not Not Cosec𝜃 2 √2 1 -1 defined √3 defined defined 2⁄ Not Not Sec𝜃 1 √2 2 -1 1 √3 defined defined Not 1⁄ Not Not Cot𝜃 √3 1 0 0 defined √3 defined defined Trigonometric identities • cos2𝜃+sin2 𝜃 =1 ⟹ sin𝜃 = √1 − 𝑐𝑜𝑠 2 𝜃 • 1+ cot2𝜃 = cosec2𝜃 ⟹ cot𝜃 = √𝑐𝑜𝑠𝑒𝑐 2 𝜃 − 1 • 1+tan2𝜃= sec2𝜃 ⟹ tan𝜃 = √𝑠𝑒𝑐 2 𝜃 − 1 Heights and Distance If O is the position of the observer (actually eye of the observer) OX is the horizontal line and A is any object above the horizontal line, OA is the line of sight. The angle between the horizontal and the line of sight. The angle between the horizontal and the line of sight. The angle between the horizontal and the line of sight. X𝑂̂A is called the angle of elevation of the object A as seen from O. When the object B is below the horizontal line, the angle X𝑂̂B i.e. the angle between the horizontal and line of sight is called the angle of depression of B as seen from O. Problems: 3 1.If sin x = , x lies in second quadrant. Then tan x=__________ 5 4 5 3 4 A) − 3 B) − 4 C) − 4 D) − 5 1 3𝜋 2.If cos x = - , 𝜋<x< then cot x=_______________ 2 2 1 2 1 2 A) B) - C) - D) √3 √3 √3 √3 𝜋 𝜋 𝜋 𝜋 𝜋 3.The values of x, if x2(sin2 4 + 2 sin2 3 ) + x(sin 6 + cos2 4 ) – cot2 6 = 0 are 3 3 3 3 A) -1 or - 2 B) 1 or - 2 C) -1 or D) 1 or 2 2 3 𝜋 5𝑐𝑜𝑠𝜃+8 𝑡𝑎𝑛𝜃 4. If tan𝜃 = - , <𝜃 <𝜋, then the value of is, 4 2 8 𝑠𝑒𝑐𝜃−3𝑐𝑜𝑠𝑒𝑐𝜃 2 5 8 A) 3 B) 8 C) 3 D) none of these 13 2𝑠𝑖𝑛∝−3𝑐𝑜𝑠∝ 5.If Sec∝ = , ∝ is acute. Then the value of is 5 4𝑠𝑖𝑛∝−9𝑐𝑜𝑠∝ 2 4 A)2 B)3 C)9 D) 3 6.The elevation of a tower 100 meters away is 30° then, the height of the tower is 100 50 1 A) meters B)100 meters C) meters D) meters √3 √3 √3 7.From a point 100 meters above the ground, the angles of depression of two objects due south on the ground are 60° and 45° then the distance between the objects is 100(√2−1) 100(√3−1) 100(√3−1) 100(√2−1) A) mts B) mts C) mts D) mts √3 √2 √3 √3 8.From a point on the line joining the feet of two poles of equal heights the angles of elevation of the are observed to be 30° and 60°. If the distance between the poles is 32 feet. Find the heights of the poles and the position of the point of observation. A) 8√3 feet B) 7√3 feet C) 5√3 feet D) none of these 𝑠𝑖𝑛65° 9.The value of is 𝑐𝑜𝑠25° A) 1 B)-1 C)0 D)not defined 10.If tan2A= cot (A-18°), where 2A is an acute angle then the value of A is, A) 18° B)36° C)60° D)90° 11.If √3 tan𝜃=1 and 𝜃 is acute, then the values of sin3𝜃 and cos2𝜃 are, 1 1 A) 1, 2 B) 2 , 1 C)0,1 D) 3,2 sin (90°−𝜃) 𝑐𝑜𝑠𝜃 12. + 1−cos(90°−𝜃) is equal to 1+𝑠𝑖𝑛𝜃 A) 2cos𝜃 B) 2sin𝜃 C)-2tan𝜃 D) 2sec𝜃 2 2 2 13.The value of sin 60°+ cos 30°- sin 45° is _________ 1 A) 1 B) sin90° C) D) Both (1) and (2) 2 2𝑠𝑖𝑛 𝐴−7𝑐𝑜𝑠𝐴 14.If 3tan A=4, then find the value of 3𝑐𝑜𝑠𝐴+4 −13 −13 29 A) B) C) ∞ D) 29 11 13 √3 𝑡𝑎𝑛𝐴−𝑐𝑜𝑡𝐴 15.If sin A= and A is an acute angle, then find the value of is 2 √3+ 𝑐𝑜𝑠𝑒𝑐𝐴 −2 2 2 A) B) C) D)-2 5 5 3+2√3 𝑎 16.If sin 𝜃 = 𝑏 , then cos𝜃 and tan𝜃 interms of a and b are √𝑏2 −𝑎2 𝑏 𝑏 𝑎 A) and B) and 𝑏 √𝑏2 −𝑎2 √𝑏2 −𝑎2 √𝑏 2 −𝑎2 √𝑎2 −𝑏2 𝑏 √𝑏2 −𝑎2 𝑎 C) and D) and 𝑎 √𝑎2 −𝑏2 𝑏 √𝑏2 −𝑎2 17.Find the measure of angle A, if (2SinA+1) (2Sin A-1) =0 A)90° B)60° C)45° D)30° 18.Find the value of 4(sin 30°+ cos 30°)-3(cos 45°+sin290°) 4 4 2 1 1 A) - 2 B) -2 C)2 D) 2 3𝑡𝑎𝑛30°−𝑡𝑎𝑛3 30° 19.The value of is _________ 1−3𝑡𝑎𝑛230° A)tan 90° B)tan60° C)tan45° D)tan30° 20.If sin 3𝜃 = 1 then 2𝜃 equal to A) 30° B) 60° C)45° D)90° 3 𝑠𝑖𝑛𝜃+𝑐𝑜𝑠𝜃−𝑡𝑎𝑛𝜃 21.If cos𝜃=5, 270°<𝜃 < 360°, then 𝑠𝑒𝑐𝜃+𝑐𝑜𝑠𝑒𝑐𝜃−𝑐𝑜𝑡𝜃 = 33 34 37 A)35 B) 35 C) 35 D)none of these 𝑐𝑜𝑡 2 30° 𝑠𝑒𝑐60° 𝑡𝑎𝑛45° 22.The value of x, if x.sin30°cos245° = 𝑐𝑜𝑠𝑒𝑐 2 45° 𝑐𝑜𝑠𝑒𝑐30° A)6 B)3 C)-3 D) -6 𝑥 𝑠𝑒𝑐 2 45° 𝑐𝑜𝑠𝑒𝑐 2 45° 23.If = 𝑐𝑜𝑡 2 30° 𝑐𝑜𝑡 2 60° , then x= 8 𝑠𝑖𝑛2 30° 𝑐𝑜𝑠2 45° 1 1 1 A) 8 B) C) D) 2 4 8 24.A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of 30°. After some time, the angle of depression becomes 60°. Find the distance travelled by the car during the time. 200 100 150 A) m B) m C) m D)60m √3 √3 √3 25.From the top of chiff 300m high the angles of depression of the top and bottom of a tower are observed to be 30° and 60°respectively, then the height of the tower is________ A) 100√2 m B)200m C)150 m D)90m nd 26. If A lies in the 2 quadrant and 3 tan A+4=0 then the value of 2cotA-5cosA+sinA is equal to −53 37 23 7 A) 10 B) 10 C) 10 D) 10 27.The value of cos1° cos2° cos3° ………. cos179° A)179 B)90 C)0 D)not defined 28.The angle of depression from the top of a tower of a point 70mts from the base is 45°. Then the height of the tower is 70 A) 70 mts B)70√2 mts C) mts D) 35mts √2 11 29.If cos ecA + cot A = , then tan A is 2 21 15 44 117 A) B) C) D) 22 16 117 43 30.If 𝑥 = 𝑟 sin 𝜃 and 𝑦 = 𝑟 cos 𝜃 then , the value of 𝑥 + 𝑦 2 is 2 1 A) 𝑟 B) 𝑟 2 C) D) 1 𝑟 31. The value of cosec 70° − sec 20° is A) 0 B) 1 C)90° D) 50° 32.If 3 𝑠𝑒𝑐𝜃 − 5 = 0, then 𝑐𝑜𝑡𝜃 is equal to 5 4 3 3 A) 3 B) 5 C) 4 D)5 33.If 𝜃 = 45°, then 𝑠𝑒𝑐𝜃 𝑐𝑜𝑡 𝜃 − 𝑐𝑜𝑠𝑒𝑐 𝜃 tan 𝜃 is A) 0 B)1 C)2√2 D)√2 34.If sin(90° − 𝜃 ) 𝑐𝑜𝑠𝜃 = 1 and 𝜃 is an acute angle then, 𝜃 is A)90° B)60° C) 30° D) none of these 35.Triangle TRY is a right angled isosceles triangle then, cos 𝑇 + cos 𝑅 + cos 𝑌 is 1 A) √2 B)2√2 C)1 + 2√2 D)1 + √2 36.The value of cot 𝜃 − sin(90° − 𝜃 ) cos(90 − 𝜃) is A) 𝑐𝑜𝑡𝜃 B) cos 2 𝜃 C) cot 2 𝜃 D) 𝑐𝑜𝑡𝜃 cos 2 𝜃 sin 𝜃 37. can also be written as √1−sin2 𝜃 sin 𝜃 A) cot 𝜃 B)√sin 𝜃 3C) D)tan 𝜃 √𝑐𝑜𝑠𝜃 13 38.If 𝑐𝑜𝑠𝑒𝑐 𝜃 = then 12 12 5 12 12 A) tan 𝜃 = B) tan 𝜃 = − 12 C) tan 𝜃 = D) tan 𝜃 = ± 5 25 25 39.𝑐𝑜𝑡𝜃 + tan 𝜃 is equal to A)𝑐𝑜𝑠𝑒𝑐 𝜃 sec 𝜃 B) sin θ sec 𝜃 C)cos 𝜃 tan 𝜃 D) sin2 𝜃 1 1 40. If sin(𝐴 − 𝐵) = and cos(𝐴 + 𝐵) = 2 then A and B will be respectively , 2 A)15° , 45° B)45°, 15° C)45°, 45° D)30°, 60° (1+tan2 𝐴) 41. (1+cot2 𝐴) = A)sec2 𝐴 B)-1 C)cot 2 𝐴 D)tan2 𝐴 cos 60°+sin 60° 42.cos 60°−sin 60° = A)−√3 + 2 B)−2 − √3 C)√3 − 2 D) None of these 12 13 sin 𝐴+5 sec 𝐴 43.If sin 𝐴 = 13 , then the value of 5 tan 𝐴 +12 𝑐𝑜𝑠𝑒𝑐 𝐴 will be A)9 B)8 C)4 D) None of these 44.The value of tan 30° sin 30° cot 60° 𝑐𝑜𝑠𝑒𝑐 30° will be 1 1 A)1 B) C) D)√3 3 √3 45.If 𝜃 increases from 0° to 90° then, sin 𝜃 changes according to : A)from −∞ to 0 B) from 0 to 1 C) from −∞ to 1 D) None of these 46.If sin 2𝐴 = cos 3𝐴, then correct statement is A) 𝐴 = 110° B)𝐴 = 30° C)𝐴 = 20° D) 𝐴 = 18° 47.If 𝛼 + 𝛽 = 90° and 𝛼 = 2𝛽, then cos 𝛼 + sin2 𝛽 is equal to 2 1 A) 1 B) 2 C) 0 D) 2 1 48.Maximum value of 𝑐𝑜𝑠𝑒𝑐 𝜃 , 0° < 𝜃 ≤ 90° is A)-1 B) 2 C)1 D)can’t be determined 2 2 49.𝑡𝑎𝑛𝑥 + 𝑠𝑖𝑛𝑥 = 𝑚 and 𝑡𝑎𝑛𝑥 − sin 𝑥 = 𝑛, then 𝑚 − 𝑛 is equal to A)4√𝑚𝑛 B) √𝑚𝑛 C) 2√𝑚𝑛 D) None of these 50.In a right angled triangle, one of the side lengths is 24 units , while the other one is 7 units. Also the angle between longest and shortest sides is known to be 𝜃. Assuming that the shortest side of triangle represents its base , value of sin 𝜃 is : 7 24 24 A) 24 B) 7 C) 25 D) data insufficient KEY ANSWERS TOPIC: TRIGONOMETRY 1.C 2.A 3.B 4.A 5.B 6.A 7.C 8.A 9.A 10.B 11.A 12.D 13.D 14.A 15.B 16.D 17.D 18.B 19.A 20.B 21.B 22.A 23.C 24.A 25.B 26.C 27.C 28.A 29.C 30.B 31.A 32.C 33.A 34.D 35.A 36.D 37.D 38.A 39.A 40.B 41.D 42.B 43.D 44.B 45.B 46.D 47.B 48.C 49.A 50.C