Question

If sin theta -cos theta = ½ then find the value of sin O+ sin O​

Answer 1

$\begin{gathered}\begin{gathered}\bf Given -  \begin{cases} &\rm \: {sin\theta \: \: - \: cos\theta \: = \dfrac{1}{2} } \\ \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf  To \:  Find :-  \begin{cases} &\sf{sin\theta \: + \: cos\theta \: }  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$1. \: \boxed{ \pink{ \rm \: {sin}^{2}\theta \: + {cos}^{2} \theta \: = 1}}$

$2. \: \boxed{ \pink{ \rm \: {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2} )}}$

$\large\underline\purple{\bold{Solution :- }}$

We know,

$\rm \: {(x + y)}^{2} + {(x - y)}^{2} = 2({x}^{2} + {y}^{2} )$

$\bf \: on \: substituting \: x \: = sin\theta \: \: and \: y \: = cos\theta \: \: we \: get$

$\rm \: {(sin\theta \: + cos\theta \: )}^{2} + {(sin\theta \: - cos\theta \: )}^{2} = 2( {sin}^{2} \theta \: + {cos}^{2} \theta \: )$

$\rm : \implies \: {(sin\theta \: + cos\theta \:)}^{2} + {(\dfrac{1}{2}) }^{2} = 2 \times 1$

$\rm : \implies \: {(sin\theta \: + cos\theta \: )}^{2} + \dfrac{1}{4} = 2$

$\rm : \implies \: {(sin\theta \: + cos\theta \: )}^{2} = 2 - \dfrac{1}{4}$

$\rm : \implies \: {(sin\theta \: + cos\theta \: )}^{2} = \dfrac{8 - 1}{4}$

$\rm : \implies \: {(sin\theta \: + cos\theta \: )}^{2} = \dfrac{7}{4}$

$\rm : \implies \: {(sin\theta \: + cos\theta \: )}^{2} = \pm \: \dfrac{ \sqrt{7} }{2}$

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Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1