Question

If cosec theta-cot theta=5 lies in quadrant

Answer 1

Answer:

The value of cosec θ - cot θ = 5 lies in second quadrant.

Step-by-step explanation:

We are given that:

cosec θ - cot θ = 5

cosec θ = 5 + cot θ

Squaring both sides:

(cosec θ)² = (5 + cot θ)²

cosec² θ = 25 + cot² θ + 10 cot θ.

Using the trigonometric identity:

1 + cot² x = cosec² x

We get:

1 + cot² θ = 25 + cot² θ + 10 cot θ.

24 + 10 cot θ = 0.

cot θ = -12 / 10 = 2.4

The value of cosec θ will be:

cosec θ = 5 + 2.4

cosec θ = 7.4

Therefore, this value lies in the second quadrant as cot θ is negative and cosec θ is positive.

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Answer 2

Concept introduction:

The plane is divided into four infinite areas, known as quadrants, by the planes of the two-dimensional Cartesian system, each of which is bordered by two semi axes. These are frequently denoted with Roman numerals and numbered from $1$ to $4:$

Given:

Here it is given that $cosec$θ$-cot$θ $=5$.

To find:

We have to find the quadrant.

Solution:

According to the question,

$cosec$θ$-cot$θ $=5$.

$cosec$θ$=5+cot$θ

$cosec^{2}$θ$=25+cot^{2}$θ$+10cot$θ

Using the trigonometric identity:

$1+cot^{2}$θ $=cosec^{2}$θ

So, we can get $cot$θ$=2.4$

Now, The value of $cosec\theta$ will be

$cosec\theta=5+2.4\\=7.4$

Therefore, this value lies in the second quadrant as $cot$ θ is negative and $cosec$θ is positive.

Final answer:

So, we have written the answer of the question and this is our final answer also.

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