Here, you will learn conditional trigonometric identities and maximum and minimum value in trigonometry.
Let’s begin –
Maximum and Minimum values in Trigonometry Expressions :
(i) 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) Minimum value of \(a^2tan^2\theta\) + \(b^2\tan^2\theta\) = 2ab where a,b > 0
(iii) -\(\sqrt{a^2 + b^2 + 2abcos(\alpha – \beta)}\) \(\le\) acos(\(\alpha + \theta\)) + bcos(\(\beta + \theta\)) \(\le\) \(\sqrt{a^2 + b^2 + 2abcos(\alpha – \beta)}\) where \(\alpha\) and \(\beta\) are known angles.
(iv) 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\(cos({\pi\over 4} — \theta)\)
Solution : We have 1 + \(sin({\pi\over 4} + \theta)\) + 2\(cos({\pi\over 4} — \theta)\)
= 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) tanA + tanB + tanC = tanA tanB tanC
(ii) cotA cotB + cotB cotC + cotC cotA = 1
(iii) \(tan{A\over 2}\) \(tan{B\over 2}\) + \(tan{B\over 2}\) \(tan{C\over 2}\) + \(tan{C\over 2}\) \(tan{A\over 2}\) = 1
(iv) \(cot{A\over 2}\) + \(cot{B\over 2}\) + \(cot{C\over 2}\) = \(cot{A\over 2}\) \(cot{B\over 2}\) \(cot{C\over 2}\)
(v) sin2A + sin2B + sin2C = 4sinA sinB sinC
(vi) cos2A + cos2B + cos2C = 1 – 4cosA cosB cosC
(vii) sinA + sinB + sinC = 4\(cos{A\over 2}\) \(cos{B\over 2}\) \(cos{C\over 2}\)
(viii) cosA + cosB + cosC = 1 + 4\(sin{A\over 2}\) \(sin{B\over 2}\) \(sin{C\over 2}\)
Some Important results :
(i) sinA sin(\(60^{\circ}\) – A) sin(\(60^{\circ}\) + A) = \(1\over 4\)sin3A
(ii) cosA cos(\(60^{\circ}\) – A) cos(\(60^{\circ}\) + A) = \(1\over 4\)cos3A
(iii) tanA tan(\(60^{\circ}\) – A) tan(\(60^{\circ}\) + A) = tan3A
(iv) cotA cot(\(60^{\circ}\) – A) cot(\(60^{\circ}\) + A) = cot3A
(v) \(sin^2A\) + \(sin^2(60^{\circ}\) – A) + \(sin^2(60^{\circ}\) + A) = \(3\over 2\)
(vi) \(cos^2A\) + \(cos^2(60^{\circ}\) – A) + \(cos^2(60^{\circ}\) + A) = \(3\over 2\)
(vii) tanA + tan(\(60^{\circ}\) + A) + tan(\(120^{\circ}\) + A) = 3tan3A
