Express sin 67 + cos 75 in terms of trigonometric ratios of angles between 0 and 45.
Solution : We have, sin 67 + cos 75 = sin(90 – 23) + cos(90 – 15) = cos 23 + sin 15
If A, B and C are the interior angles of a triangle ABC, show that sin\(B+C\over 2\) = cos\(A\over 2\).
Solution : Since A, B and C are the interior angles of a triangle ABC \(\therefore\) A + B + C = 180 \(\implies\) \(A\over 2\) + \(B\over 2\) + \(C\over 2\) = 90 \(\implies\) \(B\over 2\) + \(C\over 2\) = 90 – \(A\over 2\) \(\implies\) sin(\(B+C\over 2\)) = sin(90 – \(A\over 2\)) \(\implies\) sin\(B+C\over …
If sec 4A = cosec(A – 20), where 4A is an acute angle, find the value of A.
Solution : Given, sec 4A = cosec(A – 20) \(\implies\) cosec(90 – 4A) = cosec(A – 20) \(\implies\) 90 – 4A = A – 20 \(\implies\) 5A = 110 \(\implies\) A = 22
If tan A = cot B, prove that A + B = 90.
Solution : We have, tan A = cot B \(\implies\) tan A = tan(90 – B) \(\implies\) A = 90 – B \(\implies\) A + B = 90
If tan 2A = cot(A – 18), where 2A is an acute angle, find the value of A.
Solution : Given, tan 2A = cot(A – 18) \(\implies\) cot(90 – 2A) = cot(A – 18) \(\implies\) 90 – 2A = A – 18 \(\implies\) 3A = 108 \(\implies\) A = 36
Show that : (i) tan 48 tan 23 tan 42 tan 67 = 1 (ii) cos 38 cos 52 – sin 38 sin 52 = 0
Solution : (i) L.H.S = tan 48 tan 23 tan 42 tan 67 = \(1\over cot 48\). tan 23 tan 42 \(1\over cot 67\) = \(1\over cot (90 – 42)\). tan 23 tan 42 \(1\over cot (90 – 23)\) = \(1\over tan 42\). tan 23 tan 42 \(1\over tan 23\) = 1 = R.H.S. (ii) …
Show that : (i) tan 48 tan 23 tan 42 tan 67 = 1 (ii) cos 38 cos 52 – sin 38 sin 52 = 0 Read More »
Evaluate the following :
Question : (i) \(sin 18\over cos 72\) (ii) \(tan 26\over cot 64\) (iii) cos 48 – sin 42 (iv) cosec 31 – sec 59 Solution : (i) \(sin 18\over cos 72\) = \(sin 18\over cos (90 – 18)\) = \(sin 18\over sin 18\) = 1 (ii) \(tan 26\over cot 64\) = \(tan 26\over cot (90 …
State whether the following are true or false. Justify you answer.
Question : (i) Sin (A + B) = sin A + sin B (ii) The value of \(sin \theta\) increases as \(\theta\) increases. (iii) The value of \(cos \theta\) increases as \(\theta\) increases. (iv) \(sin \theta\) = \(cos \theta\) for all values of \(\theta\). (v) Cot A is not defined for A = 0. Solution …
State whether the following are true or false. Justify you answer. Read More »
If tan (A + B) = \(\sqrt{3}\) and tan (A – B) = \(1\over \sqrt{3}\) ; 0 < A + B \(\le\) 90 ; A \(\ge\) B, find A and B.
Solution : We have, tan (A + B) = \(\sqrt{3}\) \(\implies\) tan (A + B) = tan 60 \(\implies\) A + B = 60 …………(1) Also, tan (A – B) = \(1\over \sqrt{3}\) \(\implies\) tan(A – B) = tan 30 \(\implies\) A – B = 30 …………(2) …
Choose the correct option and justify your choice :
Question : (i) \(2 tan 30\over 1 + tan^2 30\) = (a) sin 60 (b) cos 60 (c) tan 60 (d) sin 30 (ii) \(1 – tan^2 45\over 1 + tan^2 45\) = (a) tan 90 (b) 1 (c) sin 45 (d) 0 (iii) sin 2A = 2 sin A is the true when A …
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