# 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