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nikdorinn [45]
3 years ago
14

Given sinx+1/sinx=1+cscx, find a numerical value of one trigonometric function of x.

Mathematics
1 answer:
Nana76 [90]3 years ago
3 0

Answer:

x = π/2 or

x = π/2 + 2πn (for any value of n)

Step-by-step explanation:

We need to find the numerical value of the trigonometric function sinx+1/sinx=1+cscx

Subtract 1+cscx on both sides

sinx + 1/sinx -(1+csc x) = 1+cscx - (1+cscx)

sinx + 1/sinx -1 -csc x = 0

Taking LCM i.e, sinx and solving

(sinx)(sinx) + 1 - sinx -cscx(sinx) / sinx = 0

Multiplying sinx on both sides

sin^2 x + 1 -sinx - cscx(sinx) =0

as we know, cscx = 1/ sinx

sin^2 x + 1 -sinx - (1/sinx)(sinx) = 0

Solving,

sin^2 x + 1 -sinx - 1 = 0

sin^2 x - sin x =0

Let u = sinx

Putting it in above equation

u^2 - u =0

Solving the equation:

u= 1 , u= 0

Putting back the value of u

sin(x) = 1 and sin(x) = 0

x = sin ⁻¹ (1) and x = sin ⁻¹ (1) =0

x = 90° or π/2 and x = 0 (undefined for the question)

So, x = π/2 or

x = π/2 + 2πn (for any value of n)

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\large\underline{\sf{Solution-}}

Given that,

In <u>triangle TPQ, </u>

  • RS || PQ,

  • RS = 3 cm,

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  • ar(∆ TRS) = 15 sq. cm

As it is given that, <u>RS || PQ</u>

So, it means

⇛∠TRS = ∠TPQ [ Corresponding angles ]

⇛ ∠TSR = ∠TPQ [ Corresponding angles ]

\rm\implies \: \triangle TPQ \:  \sim \: \triangle TRS \:  \:  \:  \:  \:  \:  \{AA \}

<u>Now, We know </u>

Area Ratio Theorem,

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

\rm\implies \:\dfrac{ar( \triangle \: TPQ)}{ar( \triangle \: TRS)}  = \dfrac{ {PQ}^{2} }{ {RS}^{2} }

\rm\implies \:\dfrac{ar( \triangle \: TPQ)}{15}  = \dfrac{ {6}^{2} }{ {3}^{2} }

\rm\implies \:\dfrac{ar( \triangle \: TPQ)}{15}  = \dfrac{36 }{9}

\rm\implies \:\dfrac{ar( \triangle \: TPQ)}{15}  = 4

\rm\implies \:ar( \triangle \: TPQ)  = 60 \:  {cm}^{2}

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Step-by-step explanation:

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Step-by-step explanation:

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sin(x)=\dfrac{O}{H} \ \ \ \ \ cos(x)=\dfrac{A}{H} \ \ \ \ \ tan(x)=\dfrac{O}{A}

where

  • x = angle
  • O = side opposite the angle
  • A = side adjacent the angle
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Answer:

8 in

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