Posts for Tag: Trigonometry

Trigonometric Equations – HSC Math

Whole number: n if $sin\ θ=sin\ a,$ then $θ=nπ+(-1)^n\ a$ if $cosec\ θ=cosec\ a,$ then $θ=nπ+(-1)^n\ a$ if $cos\ θ=cos\ a,$ then $θ=2nπ±a$ if $sec\ θ=sec\ a,$ then $θ=2nπ±a$ if $tan\ θ=tan\ a,$ then $θ=nπ+a$ if $cot\ θ=cot\ a,$ then $θ=nπ+a$ if $sin\ θ=0,$ then $θ=nπ$ if $sin\ θ=1,$ then $θ=(4n+1)π/2$ if $sin\ θ=-1,$ thenRead More

Inverse Trigonometric Functions – HSC Math

$sin^{-1}x+cos^{-1}x=π/2$ $tan^{-1}x+cot^{-1}x=π/2$ $sec^{-1}x+cosec^{-1}x=π/2$ $sin^{-1}(-x)=-sin^{-1}x$ $cos^{-1}(-x)=π-cos^{-1}x$ $tan^{-1}(-x)=tan^{-1}x$ $cot^{-1}(-x)=π-cot^{-1}x$ $sec^{-1}(-x)=π-sec^{-1}x$ $cosec^{-1}(-x)=-cosec^{-1}x$ $sin^{-1}(1/x)=cosec^{-1}x$ $cos^{-1}(1/x)=sec^{-1}x$ $tan^{-1}(1/x)=cot^{-1}x$ $sin^{-1}x+sin^{-1}y=sin^{-1}(x√{1-y^2}+y√{1-x^2})$ $sin^{-1}x-sin^{-1}y=sin^{-1}(x√{1-y^2}-y√{1-x^2})$ $cos^{-1}x+cos^{-1}y=cos^{-1}(xy-√{1-x^2}√{1-y^2})$ $cos^{-1}x-cos^{-1}y=cos^{-1}(xy+√{1-x^2}√{1-y^2})$ $tan^{-1}x+tan^{-1}y=tan^{-1}{x+y}/{1-xy}$ $tan^{-1}x-tan^{-1}y=tan^{-1}{x-y}/{1+xy}$ $2sin^{-1}(x)=sin^{-1}(2x√{1-x^2})$ $2cos^{-1}(x)=cos^{-1}(2x^2-1)$ $2tan^{-1}(x)=tan^{-1}{2x}/{1-x^2}=cos^{-1}{1-x^2}/{1+x^2}$ $3sin^{-1}(x)=sin^{-1}(3x-4x^3)$ $3cos^{-1}(x)=cos^{-1}(4x^3-3x)$ $3tan^{-1}(x)=tan^{-1}{3x-x^3}/{1-3x^2}$ $1/2sin^{-1}(x)=tan^{-1}{1-√{1-x^2}}/x$ $1/2cos^{-1}(x)=cos^{-1}√{{1+x}/2}=sin^{-1}√{{1-x}/2}=tan^{-1}√{{1-x}/{1+x}}$ $1/2tan^{-1}(x)=tan^{-1}{√{1+x^2}-1}/x$ $sin^{-1}(x)=cos^{-1}√{1-x^2}$ $=tan^{-1}x/√{1-x^2}$ $=cot^{-1}√{1-x^2}/x$ $=sec^{-1}1/√{1-x^2}$ $=cosec^{-1}1/x$

Trigonometric Properties of a Triangle – HSC Math

Sine Rule $a/{Sin\ A}=b/{Sin\ B}=c/{Sin\ C}$ Cos Rule $cos\ A={b^2+c^2-a^2}/{2bc}$ $cos\ B={a^2+c^2-b^2}/{2ac}$ $cos\ C={a^2+b^2-c^2}/{2ab}$ $a=b\ cos\ C+c\ cos\ B$ $b=c\ cos\ A+a\ cos\ C$ $c=a\ cos\ B+b\ cos\ A$ Tan Rule $tan\ {B-C}/2={b-c}/{b+c}\ cot\ A/2$ $tan\ {A-B}/2={a-b}/{a+b}\ cot\ C/2$ $tan\ {C-A}/2={c-a}/{c+a}\ cot\ B/2$ $tan\ A/2=√{(s-b)(s-c)}/√{s(s-a)}={(s-b)(s-c)}/{∆}$ $tan\ B/2=√{(s-a)(s-c)}/√{s(s-c)}={(s-a)(s-c)}/{∆}$ $tan\ C/2=√{(s-a)(s-b)}/√{s(s-c)}={(s-a)(s-b)}/{∆}$ Here, $s={a+b+c}/2, ∆=$Area of that triangle.Read More

Trigonometric Formulas – HSC Math

In circular system, $θ°={\arc}/{\radius}=s/r$ or, $s=rθ$ $1 \radian=1 \rad=1^c=(180/π)^°$ Area of a circular sector (symbol ⌔) $⌔\AOB=1/2r^2θ$ [θ in radian] $sin\ θ=1/{cosec\ θ}; cos\ θ=1/{sec\ θ}; tan\ θ=1/{cot\ θ}$ $tan\ θ={sin \θ}/{cos\ θ}; cot\ θ={cos\ θ}/{sin\ θ}$ $sin^2\ θ+cos^2\ θ=1$ $sec^2\ θ-tan^2\ θ=1$ $cosec^2\ θ-cot^2\ θ=1$ $-1≤sin\ θ≤1$ $-1≤cos\ θ≤1$ $cosec\ θ≤-1, 1≤cosec\ θ$ $sec\Read More

Relations to Hyperbolic Functions

Imaginary unit: i $sin(ix)=i\ sinhx$ $tan(ix)=i\ tanhx$ $cot(ix)=-i\ cothx$ $sec(ix)=sechx$ $cosec(ix)=-i\ cosechx$

Trigonometric Equations

Whole number: n if $sin\ x=a,$ then $x=πn+(-1)^n sin^{-1}\ a$ if $sin\ x=sin\ a,$ then $x=πn+(-1)^n\ a$ if $cos\ x=a,$ then $x=2πn±cos^{-1}\ a$ if $cos\ x=cos\ a,$ then $x=2πn±a$ if $tan\ x=a,$ then $x=πn+tan^{-1}\ a$ if $tan\ x=tan\ a,$ then $x=πn+a$ if $cot\ x=a,$ then $x=πn+cot^{-1}\ a$ if $cot\ x=cot\ a,$ then $x=πn+a$ if $sec\Read More

Relations between Inverse Trigonometric Functions

$sin^{-1}(-x)=-sin^{-1}x$ $sin^{-1}x=π/2-cos^{-1}x$ $sin^{-1}x=cos^{-1}√{1-x^2}, 0≤x≤1$ $sin^{-1}x=-cos^{-1}√{1-x^2}, -1≤x≤0$ $sin^{-1}x=tan^{-1}x/√{1-x^2}, x^2

Power of Trigonometric Functions

$sin^2θ={1-cos\ 2θ}/2 $ $sin^3θ={3sinθ-sin3θ}/4$ $sin^4θ={cos4θ-4cos2θ+3}/8$ $sin^5θ={10sinθ-5sin3θ+sin5θ}/16 $ $sin^6θ={10-15cos2θ+6cos4θ-cos6θ}/32$ $cos^2θ={1+cos\ 2θ}/2 $ $cos^3θ={3cosθ+cos3θ}/4$ $cos^4θ={cos4θ+4cos2θ+3}/8$ $cos^5θ={10cosθ+5sin3θ+cos5θ}/16 $ $cos^6θ={10+15cos2θ+6cos4θ+cos6θ}/32$

Trigonometric Expression to Sum Transformation

$sinθ\ sinβ={cos(θ-β)-cos(θ+β)}/2$ $cosθ\ cosβ={cos(θ-β)+cos(θ+β)}/2$ $sinθ\ cosβ={sin(θ-β)+sin(θ+β)}/2$ $tanθ\ tanβ={tanθ+tanβ}/{cotθ+cotβ}$ $cotθ\ cotβ={cotθ+cotβ}/{tanθ+tanβ}$ $tanθ\ cotβ={tanθ+cotβ}/{cotθ+tanβ}$

Trigonometric Expression to Product Transformation

$sinθ+sinβ=2sin{θ+β}/2cos{θ-β}/2$ $sinθ-sinβ=2cos{θ+β}/2sin{θ-β}/2$ $cosθ+cosβ=2cos{θ+β}/2cos{θ-β}/2$ $cosθ-cosβ=-2sin{θ+β}/2sin{θ-β}/2$ $tanθ+tanβ={sin(θ+β)}/{cosθ\ cosβ}$ $tanθ-tanβ={sin(θ-β)}/{cosθ\ cosβ}$ $cotθ+cotβ={sin(β+θ)}/{sinθ\ sinβ}$ $cotθ-cotβ={sin(β-θ)}/{sinθ\ sinβ}$ $cosθ+sinθ=√2cos{π/4-θ)}=√2sin{π/4+θ)}$ $cosθ-sinθ=√2sin{π/4-θ)}=√2cos{π/4+θ)}$ $tanθ+cotβ={cos(θ-β)}/{cosθ\ sinβ}$ $tanθ-cotβ={cos(θ+β)}/{cosθ\ sinβ}$ $1+cosθ=2cos^2{θ/2}$ $1-cosθ=2sin^2{θ/2}$ $1+sinθ=2cos^2(π/4-θ/2)$ $1-sinθ=2sin^2(π/4-θ/2)$