Question 1.<\/strong>
\nIn \u0394 ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :
\n(i) sin A, cos A
\n(ii) sin C, cos C<\/p>\nSolution:
\n<\/strong>In \u2206ABC by applying Pythagoras theorem
\nAC2<\/sup>\u00a0= AB2<\/sup>\u00a0+ BC2<\/sup>
\n= (24)2<\/sup>\u00a0+ (7)2<\/sup>
\n= 576 + 49
\n= 625
\nAC = \u221a625\u00a0= 25 cm
\n<\/p>\nQuestion 2.<\/strong> In Figure, find tan P \u2013 cot R.
\n
\nSolution:
\n<\/strong>In\u00a0 \u2206PQR by applying Pythagoras theorem
\nPR2<\/sup> = PQ2<\/sup> + QR2<\/sup>
\n(13)2<\/sup> = (12)2<\/sup> + QR2<\/sup>
\n169 = 144 + QR2<\/sup>
\n25 = QR2<\/sup>
\nQR = 5
\n<\/p>\nQuestion 3.<\/strong> If sin A =3\/4, calculate cos A and tan A.<\/p>\nSolution:
\n<\/strong>Let\u00a0 \u2206ABC be a right angled triangle, right angled at point B.
\nGiven that
\n
\nsin A = 3\/4
\nBC\/AC = 3\/4
\nLet BC be 3 K so AC will be 4 K where K \u00a1s a positive integer.
\nNow applying Pythagoras theorem in \u2206ABC
\nAC2<\/sup> = AB2<\/sup>\u00a0+ BC2<\/sup>
\n(4 K)2<\/sup> = AB2<\/sup>\u00a0+ (3 K)2<\/sup>
\n16 K2<\/sup> – g K2<\/sup> = AB2<\/sup>
\n7 K2<\/sup> = AB2<\/sup>
\nAB =\u00a0\u221a7k
\n<\/p>\nQuestion 4.<\/strong>
\nGiven 15 cot A = 8, find sin A and sec A.<\/p>\nSolution:
\n<\/strong>Consider a right triangle, right angled at B
\n
\nLet AB be 8 K so BC will be 15 K where K is a positive integer.
\nNow applying Pythagoras theorem in \u2206ABC
\nAC2<\/sup> = AB2<\/sup> + BC2<\/sup>
\n= (8K)2<\/sup> + (15K)2<\/sup>
\n= 64 K2<\/sup> + 225 K2<\/sup>
\n= 289 K2<\/sup>
\nAC = 17 K
\n<\/strong><\/p>\nQuestion 5.<\/strong> Given sec \u03b8 = 13\/12, calculate all other trigonometric ratios.<\/p>\nSolution:
\n<\/strong>Consider a right angle triangle\u00a0\u2206ABC right angled at point B.
\n<\/strong>
\n<\/p>\nQuestion 6.<\/strong>
\nIf \u2220A and \u2220B are acute angles such that cos A = cos B, then show that \u2220A = \u2220B.<\/p>\nSolution:
\n<\/strong>Since cos A = cos B
\nSo AB\/AC = BC\/AC
\nAB = BC
\nSo \u2220A = \u2220B (Angles opposite to equal sides are equal in length)<\/p>\nQuestion 7.<\/strong> If cot \u03b8 =7\/8, evaluate :
\n(i) \\(\\frac { \\left( 1+sin\\theta \\right) \\left( 1-sin\\theta \\right) }{ \\left( 1+cos\\theta \\right) \\left( 1-cos\\theta \\right) } \\)
\n(ii) Cot2<\/sup>\u03b8<\/p>\nSolution:
\n<\/strong>Consider a right angle triangle\u00a0\u2206ABC right angled at point B.
\n
\n<\/strong><\/p>\nQuestion 8.<\/strong> If 3cot A = 4\/3 , check whether \\(\\frac { 1-tan^{ 2 }A }{ 1+tan^{ 2 }A } ={ cos }^{ 2 }A-{ sin }^{ 2 }A\\)\u00a0or not.<\/p>\nSolution:
\n
\n<\/strong><\/p>\nQuestion 9.<\/strong> In triangle ABC, right-angled at B, if tan A =1\/\u221a3 find the value of:
\n(i) sin A cos C + cos A sin C
\n(ii) cos A cos C \u2013 sin A sin C<\/p>\nSolution:
\n
\n<\/strong><\/p>\nQuestion 10.<\/strong> In \u0394 PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.<\/p>\nSolution:
\n
\n<\/strong><\/p>\nQuestion 11.<\/strong> State whether the following are true or false. Justify your answer.
\n(i) The value of tan A is always less than 1.
\n(ii) sec A = 12\/5 for some value of angle A.
\n(iii) cos A is the abbreviation used for the cosecant of angle A.
\n(iv) cot A is the product of cot and A.
\n(v) sin \u03b8 = 4\/3 for some angle \u03b8.<\/p>\nSolution:
\n<\/strong>
\nLet AC be 12 K, AB will be 5 K, where K is a positive integer
\nNow applying Pythagoras theorem in \u2206ABC
\nAC2<\/sup> = AB2<\/sup> + BC2<\/sup>
\n(12 K)2<\/sup> = (5 K)2<\/sup> + BC2<\/sup>
\n144 K2<\/sup> = 25 K2<\/sup>\u00a0+ BC2<\/sup>
\nBC2<\/sup> = 119 K2<\/sup>
\nBC = 109 K
\nWe may observe that for given two sides AC = 12 K and AB = S K
\nBC should be such that –
\nAC – AB < BC < AC + AB
\n12 K – 5 K < BC <12 K + 5 K
\n7 K < BC < 17 K
\nBut BC = 10.9 K.
\nClearly such a triangle is possible and hence such value of sec A is possible. Hence, the given statement is true.<\/p>\n(iii) Abbreviation used for cosecant of angle A is cosec A. And cos A is the abbreviation used for cosine of angle A. Hence, the given statement is false.<\/p>\n
(iv) cot A is not the product of cot and A but it is cotangent of \u2220A. Hence, the given statement is false.<\/p>\n
(v) sine \u03b8 = 4\/3
\nWe know that in a right angle triangle
\n
\nIn a right angle triangle hypotenuse is always greater then the remaining two sides.
\nHence such value of sin\u00a0\u03b8 is not possible. Hence, the given statement is false.<\/p>\n
We hope the NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1 help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Ex 8.1, drop a comment below and we will get back to you at the earliest.<\/p>\n