Conditional Trigonometric Identities – Maximum & Minimum Value

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