Graph of Trigonometric Functions – Domain & Range

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

Graph of Trigonometric Functions :

y = sinx

y = cosx


y = tanx

y = cotx


y = secx

y = cosecx


Values of T-Ratio of some standard angles

Angles
T-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\(36^{\circ}\)

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

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