Here, you will learn conditional trigonometric identities and maximum and minimum value in trigonometry.
Letโs begin โ
Maximum and Minimum values in Trigonometry Expressions :
(i)ย acos\(\theta\) + bcos\(\theta\) will always lie in the interval [-\(\sqrt{a^2+b^2}\), \(\sqrt{a^2+b^2}\)] i.e. the maximum and minimum values are \(\sqrt{a^2+b^2}\), -\(\sqrt{a^2+b^2}\) respectively.
(ii)ย Minimum value of \(a^2tan^2\theta\) + \(b^2\tan^2\theta\) = 2ab where a,b > 0
(iii)ย -\(\sqrt{a^2 + b^2 + 2abcos(\alpha โ \beta)}\) \(\le\) acos(\(\alpha + \theta\)) + bcos(\(\beta + \theta\)) \(\le\) \(\sqrt{a^2 + b^2 + 2abcos(\alpha โ \beta)}\) where \(\alpha\) and \(\beta\) are known angles.
(iv)ย In case a quadratic in sin\(\theta\) & cos\(\theta\) is given then the maximum and minimum values can be obtained by making perfect square.
Example : Find the maximum value of 1 + \(sin({\pi\over 4} + \theta)\) + 2\(cos({\pi\over 4} โ \theta)\)
Solution : We have 1 + \(sin({\pi\over 4} + \theta)\) + 2\(cos({\pi\over 4} โ \theta)\)
= 1 + \(1\over sqrt{2}\)(cos\(\theta\) + sin\(\theta\)) + \(\sqrt{2}\)(cos\(\theta\) + sin\(\theta\)) = 1 + (\({1\over \sqrt{2}} + \sqrt{2}\)) (cos\(\theta\) + sin\(\theta\))
= 1 + (\({1\over \sqrt{2}} + \sqrt{2}\)) . \(\sqrt{2}\) = 4
Conditional Trigonometric Identities :
If A + B + C = \(180^{\circ}\),then
(i)ย tanA + tanB + tanC = tanA tanB tanC
(ii)ย cotA cotB + cotB cotC + cotC cotA = 1
(iii)ย \(tan{A\over 2}\) \(tan{B\over 2}\) + \(tan{B\over 2}\) \(tan{C\over 2}\) + \(tan{C\over 2}\) \(tan{A\over 2}\) = 1
(iv)ย \(cot{A\over 2}\) + \(cot{B\over 2}\) + \(cot{C\over 2}\) = \(cot{A\over 2}\) \(cot{B\over 2}\) \(cot{C\over 2}\)
(v)ย ย sin2A + sin2B + sin2C = 4sinA sinB sinC
(vi)ย cos2A + cos2B + cos2C = 1 โ 4cosA cosB cosC
(vii)ย sinA + sinB + sinC = 4\(cos{A\over 2}\) \(cos{B\over 2}\) \(cos{C\over 2}\)
(viii)ย cosA + cosB + cosC = 1 + 4\(sin{A\over 2}\) \(sin{B\over 2}\) \(sin{C\over 2}\)
Some Important results :
(i)ย sinA sin(\(60^{\circ}\) โ A) sin(\(60^{\circ}\) + A) = \(1\over 4\)sin3A
(ii)ย cosA cos(\(60^{\circ}\) โ A) cos(\(60^{\circ}\) + A) = \(1\over 4\)cos3A
(iii)ย tanA tan(\(60^{\circ}\) โ A) tan(\(60^{\circ}\) + A) = tan3A
(iv)ย cotA cot(\(60^{\circ}\) โ A) cot(\(60^{\circ}\) + A) = cot3A
(v)ย \(sin^2A\) + \(sin^2(60^{\circ}\) โ A) + \(sin^2(60^{\circ}\) + A) = \(3\over 2\)
(vi)ย \(cos^2A\) + \(cos^2(60^{\circ}\) โ A) + \(cos^2(60^{\circ}\) + A) = \(3\over 2\)
(vii)ย tanA + tan(\(60^{\circ}\) + A) + tan(\(120^{\circ}\) + A) = 3tan3A
