Posts for Tag: Trigonometry

Half Angle Tangent Identities

$sinθ={2tanθ/2}/{1+tan^2θ/2}$ $cosθ={1-tan^2θ/2}/{1+tan^2θ/2}$ $tanθ={2tanθ/2}/{1-tan^2θ/2}$ $cotθ={1-tan^2θ/2}/{2tanθ/2}$

Trigonometric Half Angle Formulas

$sinθ/2=±√{{1-cosθ}/2}$ $cosθ/2=±√{{1+cosθ}/2}$ $tanθ/2=±√{{1-cosθ}/{1+cosθ}}={sinθ}/{1+cosθ}$ $={1-cosθ}/{sinθ}=cosecθ-cotθ$ $cotθ/2=±√{{1+cosθ}/{1-cosθ}}={sinθ}/{1-cosθ}$ $={1+cosθ}/{sinθ}=cosecθ+cotθ$

Trigonometric Multiple Angle Formulas

$sin\ 3θ=3sinθ-4sin^3θ=3cos^2θ\ sinθ-sin^3θ$ $sin\ 4θ=4sinθ\ cosθ-8sin^3θ\ cosθ$ $sin\ 5θ=5sinθ-20sin^3θ+16sin^5θ$ $cos\ 3θ=4cos^3θ-3cosθ=cos^3θ-3cosθ\ sin^2θ$ $cos\ 4θ=8cos^4θ-8cos^2θ+1$ $cos\ 5θ=16cos^5θ-20cos^3θ+5cosθ$ $tan\ 3θ={3tanθ-tan^3θ}/{1-3tan^2θ}$ $tan\ 4θ={4tanθ-4tan^3θ}/{1-6tan^2θ+tan^4θ}$ $tan\ 5θ={tan^5θ-10tan^3θ+5tanθ}/{1-10tan^2θ+5tan^4θ}$ $cot\ 3θ={cot^3θ-3cotθ}/{3cot^2θ-1}$ $cot\ 4θ={1-6tan^2θ+tan^4θ}/{4tanθ-4tan^3θ}$ $cot\ 5θ={1-10tan^2θ+5tan^4θ}/{tan^5θ-10tan^3θ+5tanθ}$

Trigonometric Double Angle Formulas

$sin\ 2θ=2sinθ\ cosθ={2tanθ}/{1+tan^2θ}$ $cos\ 2θ=cos^2θ-sin^2θ=1-2sin^2θ=2cos^2θ-1={1-tan^2θ}/{1+tan^2θ}$ $tan\ 2θ={2tanθ}/{1-tan^2θ}=2/{cotθ-tanθ}$ $cot\ 2θ={cot^2θ-1}/{2cotθ}={cotθ-tanθ}/2$

Trigonometric Addition and Subtraction Formulas

$sin\ (α+β)=sin\ α.cos\ β+sin\ β\ cos\ α$ $sin\ (α-β)=sin\ α.cos\ β-sin\ β\ cos\ α$ $cos\ (α+β)=cos\ α.cos\ β-sin\ α\ sin\ β$ $cos\ (α-β)=cos\ α.cos\ β+sin\ α\ sin\ β$ $tan\ (α+β)={tan\ α+tan\ β}/{1-tan\ α\ tan\ β}$ $tan\ (α-β)={tan\ α-tan\ β}/{1+tan\ α\ tan\ β}$ $cot\ (α+β)={1-tan\ α\ tan\ β}/{tan\ α+tan\ β}$ $cot\ (α-β)={1+tan\ α\ tan\ β}/{tan\ α-tan\Read More

Relation between Trigonometric Functions

$sin\ θ=1/{cosec\ θ}=±√{1-cos^2θ}=±√{1/2(1-cos\ 2θ)}$ $=2cos^2(θ/2-π/4)-1={2tan{θ/2}}/{1+tan^2{θ/2}}$ $cos\ θ=1/{sec\ θ}=±√{1-sin^2θ}=±√{1/2(1+cos\ 2θ)}$ $=2cos^2{θ/2}-1={1-tan^2{θ/2}}/{1+tan^2{θ/2}}$ $tan\ θ=1/{cot\ θ}={sin\ θ}/{cos\ θ}=±√{sec^2\ θ -1}$ $={sin\ 2θ}/{1+cos\ 2θ}={1-cos\ 2θ}/{sin\ 2θ}$ $=±√{{1-cos\ 2θ}/{1+cos\ 2θ}}={2tan\ {θ/2}}/{1+tan^2{θ/2}}$ $cot\ θ=1/{tan\ θ}={cos\ θ}/{sin\ θ}=±√{cosec^2\ θ -1}$ $={1+cos\ 2θ}/{cos\ 2θ}={sin\ 2θ}/{1-cos\ 2θ}$ $=±√{{1+cos\ 2θ}/{1-cos\ 2θ}}={1+tan^2{θ/2}}/{2tan\ {θ/2}}$ $sec\ θ=1/{cos\ θ}=±√{1+tan^2θ}={1+tan^2{θ/2}}/{1-tan^2{θ/2}}$ $cosec\ θ=1/{sin\ θ}=±√{1+cot^2θ}={1+tan^2{θ/2}}/{2tan{θ/2}}$

Periodicity of Trigonometric Functions

$sin(θ±2πn)=sin\ θ, \period\ 2π \or\ 360°$ $cos(θ±2πn)=cos\ θ, \period\ 2π \or\ 360°$ $tan(θ±πn)=tan\ θ, \period\ π \or\ 180°$ $cot(θ±πn)=cot\ θ, \period\ π \or\ 180°$ $sec(θ±2πn)=sec\ θ, \period\ 2π \or\ 360°$ $cosec(θ±2πn)=cosec\ θ, \period\ 2π \or\ 360°$

Trigonometric Functions of Common Angles

Trigonometric functions of some common angles: 0°, 30°, 45°, 60°, 90°, 120°, 180°, 270°, 360°. Trigonometric functions of some other angles: 15°, 18°, 36°, 54°, 72°, 75°.

Important Trigonometric Formulas

$sin^2θ+cos^2θ=1$ $sec^2θ-tan^2θ=1$ $cosec^2θ-cot^2θ=1$ $tan\ θ={sin\ θ}/{cos\ θ}$ $cot\ θ={cos\ θ}/{sin\ θ}$ $tan\ θ.cot\ θ=1$ $sin\ θ=1/{cosec\ θ}$ $cos\ θ=1/{sec\ θ}$ $tan\ θ=1/{cot\ θ}$ $cot\ θ=1/{tan\ θ}$ $sec\ θ=1/{cos\ θ}$ $cosec\ θ=1/{sin\ θ}$

Trigonometric Functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. $sin\ θ=y/r={BC}/{AB}$ $cos\ θ=x/r={AC}/{AB}$ $tan\ θ=y/x={BC}/{AC}$ $cot\ θ=x/y={AC}/{BC}$ $sec\ θ=r/x={AB}/{AC}$ $cosec\ θ=r/y={AB}/{BC}$ Sine function: $y=sin\ θ, -1≤sin\ θ≤1$ Cosine function: $y=cos\ θ, -1≤cos\Read More