# Graph of Trigonometric Functions – Domain & Range

Here, you will learn graph of trigonometric functions and domain & range of trigonometric functions.

y = sinx

y = cosx

y = tanx

y = cotx

y = secx

y = cosecx

## Values of T-Ratio of some standard angles

 AnglesT-Ratio 0 $$\pi\over 6$$ $$\pi\over 4$$ $$\pi\over 3$$ $$\pi\over 2$$ $$\pi$$ $$sin\theta$$ 0 $$1\over 2$$ $$1\over \sqrt{2}$$ $$\sqrt{3}\over 2$$ 1 0 $$cos\theta$$ 1 $$\sqrt{3}\over 2$$ $$1\over \sqrt{2}$$ $$1\over 2$$ 0 -1 $$tan\theta$$ 0 $$1\over \sqrt{3}$$ 1 $$\sqrt{3}$$ N.D 0 $$cot\theta$$ N.D $$\sqrt{3}$$ 1 $$1\over \sqrt{3}$$ 0 N.D $$sec\theta$$ 1 $$2\over \sqrt{3}$$ $$\sqrt{2}$$ 2 N.D -1 $$cosec\theta$$ N.D 2 $$\sqrt{2}$$ $$2\over \sqrt{3}$$ 1 N.D

N.D = Not defined

## Domain, Ranges and Periodicity of Trigonometric function

 T-Ratio Domain Range Period sin x R [-1, 1] $$2\pi$$ cos x R [-1, 1] $$2\pi$$ tan x R – {(2n+1)$$\pi/2$$; n $$\in$$ I} R $$\pi$$ cot x R – {n$$\pi$$ : n $$\in$$ I} R $$\pi$$ sec x R – {(2n+1)$$\pi/2$$; n $$\in$$ I} (-$$\infty$$, -1] $$\cup$$ [1, $$\infty$$] $$2\pi$$ cosec x R – {n$$\pi$$ : n $$\in$$ I} (-$$\infty$$, -1] $$\cup$$ [1, $$\infty$$] $$2\pi$$

## Trigonometric ratios of some standard angles :

(i)  sin$$18^{\circ}$$ = sin$$\pi\over 10$$ = $$\sqrt{5}-1\over 4$$ = cos$$72^{\circ}$$ = cos$$2\pi\over 5$$

(ii)  cos$$36^{\circ}$$ = cos$$\pi\over 5$$ = $$\sqrt{5}+1\over 4$$ = sin$$54^{\circ}$$ = sin$$3\pi\over 10$$

(iii)  sin$$72^{\circ}$$ = sin$$2\pi\over 5$$ = $$\sqrt{10 + 2\sqrt{5}}\over 4$$ = cos$$18^{\circ}$$ = cos$$\pi\over 10$$

(iv)  sin$$36^{\circ}$$ = sin$$\pi\over 5$$ = $$\sqrt{10 – 2\sqrt{5}}\over 4$$ = cos$$54^{\circ}$$ = cos$$3\pi\over 10$$

(v)  sin$$15^{\circ}$$ = sin$$\pi\over 12$$ = $$\sqrt{3}-1\over {2\sqrt{2}}$$ = cos$$75^{\circ}$$ = cos$$5\pi\over 12$$

(vi)  cos$$15^{\circ}$$ = sin$$\pi\over 12$$ = $$\sqrt{3}+1\over {2\sqrt{2}}$$ = sin$$75^{\circ}$$ = sin$$5\pi\over 12$$

(vii)  tan$$15^{\circ}$$ = tan$$\pi\over 12$$ = $$2 – \sqrt{3}$$ = $$\sqrt{3}-1\over {\sqrt{3}+1}$$ = cot$$75^{\circ}$$ = cot$$5\pi\over 12$$

(viii)  tan$$75^{\circ}$$ = tan$$5\pi\over 12$$ = $$2 + \sqrt{3}$$ = $$\sqrt{3}+1\over {\sqrt{3}-1}$$ = cot$$15^{\circ}$$ = cot$$\pi\over 12$$

(ix)  tan($$22.5^{\circ}$$) = tan$$\pi\over 8$$ = $$\sqrt{2}-1$$ = cot($$67.5^{\circ}$$) = cot$$3\pi\over 8$$

(x)  tan($$67.5^{\circ}$$) = tan$$3\pi\over 8$$ = $$\sqrt{2}+1$$ = cot($$22.5^{\circ}$$) = cot$$\pi\over 8$$

Example : Evaluate sin$$78^{\circ}$$ – sin$$66^{\circ}$$ – sin$$42^{\circ}$$ + sin$$6^{\circ}$$

Solution : (sin$$78^{\circ}$$ – sin$$66^{\circ}$$) – (sin$$42^{\circ}$$ – sin$$6^{\circ}$$-)

= 2cos($$60^{\circ}$$)sin($$18^{\circ}$$) – 2cos($$36^{\circ}$$)sin($$30^{\circ}$$)

= sin$$18^{\circ}$$ – cos$$36^{\circ}$$

= ($$\sqrt{5}-1\over 4$$) – ($$\sqrt{5}+1\over 4$$) = $$-1\over 2$$