## Trigonometric formulas and identities for class 10

The six trigonometric ratios
 $1. \space sin \spaceθ = {Side \space opposite \space to \spaceθ\over Hypotenuse}$ $2. \space cos \spaceθ = {Side \space adjacent \space to \spaceθ\over Hypotenuse}$ $3. \space tan \spaceθ = {Side \space opposite \space to \spaceθ\over \space Side \space adjacent \space to \spaceθ}$ $4. \space cosecant \spaceθ = {1\over sin\spaceθ}$ $5. \space secant \spaceθ = {1\over cos \spaceθ}$ $6. \space cot \spaceθ = {1\over tan \spaceθ}$
Trigonometric ratios of some specific angles
 Fig.1 Fig.2 Fig.3
Specific angles
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Trigonometric ratios
${θ \space = \space 0°}$ ${θ \space = \space 30°}$ ${θ \space = \space 45°}$ ${θ \space = \space 60°}$ ${θ \space = \space 90°}$
${sin \space θ}$ ${0}$ ${1\over 2}$ ${1\over \sqrt{2}}$ ${\sqrt{3} \over 2}$ ${1}$
${cos \space θ}$ ${1}$ ${\sqrt{3} \over 2}$ ${1\over \sqrt{2}}$ ${1\over 2}$ ${0}$
${tan \space θ}$ ${0}$ ${1\over \sqrt{3}}$ ${1}$ ${\sqrt{3}}$ ${Not\space defined}$
${cosec \space θ}$ ${Not\space defined}$ ${2}$ ${\sqrt{2}}$ ${2\over \sqrt{3}}$ ${1}$
${sec \space θ}$ ${1}$ ${2\over \sqrt{3}}$ ${\sqrt{2}}$ ${2}$ ${Not\space defined}$
${cot \space θ}$ ${Not\space defined}$ ${\sqrt{3}}$ ${1}$ ${1\over \sqrt{3}}$ ${0}$
Trigonometric ratios of complementary angles
Angle $$θ$$ and $$(90° - θ$$)are complementary angles
 ${sin \space θ = \space cos \space (90°-θ)}$ ${cos \space θ = \space sin \space (90°-θ)}$ ${tan \space θ = \space cot \space (90°-θ)}$ ${cosec \space θ = \space sec \space (90°-θ)}$ ${sec \space θ = \space cosec \space (90°-θ)}$ ${cot \space θ = \space tan \space (90°-θ)}$
Trigonometric identities
 ${sin^2 \space θ + \space cos^2 \space θ \space= \space 1}$ ${1+tan^2 \space θ = \space sec^2\space θ}$ ${1+cot^2 \space θ = \space cosec^2\space θ}$
Alternate forms 1:
 ${cos^2 \space θ = \space 1 - \space sin^2 \space θ}$ ${sec^2 \space θ - \space tan^2 \space θ \space = \space 1}$ ${cosec^2 \space θ - \space cot^2 \space θ \space = \space 1}$ ${sin^2 \space θ = \space 1 - \space cos^2 \space θ}$ ${sec^2 \space θ - \space 1 \space = \space tan^2 \space θ}$ ${cosec^2 \space θ - \space 1 \space = \space cot^2 \space θ}$
Alternate forms 2:
 $$sin \space θ$$ in terms of $$cos \space θ, \space tan \spaceθ$$ $sin θ = {\sqrt{1-cos^2θ}}$ $sin θ = {tanθ \over \sqrt{1-tan^2θ}}$ $$cos \space θ$$ in terms of $$sin \space θ, \space tan \spaceθ$$ $cos θ = {\sqrt{1-sin^2θ}}$ $cos θ = {1\over \sqrt{1+tan^2θ}}$ $$tan \space θ$$ in terms of $$sin \space θ, \space cos \space θ$$ $tan θ = {sin θ \over \sqrt{1-sin^2θ}}$ $cos θ = {\sqrt{1+cos^2θ} \over cosθ }$