App for CBSE class 10, Karnataka SSLC, and BSc Nursing

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
Specific angles
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 \(sin \space θ, \space cos \space θ\) \[tan θ = {sin θ \over \sqrt{1-sin^2θ}}\] \[cos θ = {\sqrt{1+cos^2θ} \over cosθ }\]
To know more about similar topics download Akshara app.