Question

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.Solution:​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\sf{sin\: A=\dfrac{1}{\sqrt{1+{cot}^{2}A}}}$

$\sf{sec\: A=\dfrac{\sqrt{{cot}^{2}A+1}}{cot\: A}}$

$\sf{tan\: A=\dfrac{1}{cot\: A}}$

Step-by-step explanation:

We've been asked to calculate the sin A, sec A and tan A in terms of cot A.

Let us firstly calculate sin A, in terms of cot A.

We know, cosec²A is the sum of one and cot² A. So, in order to find the value of sin A in terms of cot A, firstly we need to calculate the value of cosec A.

$\twoheadrightarrow\quad\sf{{cosec}^{2}A=1+{cot}^{2}A}$

Transposing the square from LHS to RHS,

$\twoheadrightarrow\quad\sf{cosec\: A=\sqrt{1+{cot}^{2}A}}$

We've calculated the value of cosec A. As, we know, sin A equals to the reciprocal of cosec A.

$\twoheadrightarrow\quad\sf{sin\: A=\dfrac{1}{cosec\: A}}$

Substituting value of cosec A.

$\twoheadrightarrow\quad\bold{sin\: A=\dfrac{1}{\sqrt{1+{cot}^{2}A}}}$

Therefore, the value of sin A in terms of cot A is, 1/√(1+cot²A)

Now, let us calculate the value of sec A. We know, sec²A equals to the sum of one and tan²A.

$\twoheadrightarrow\quad\sf{{sec}^{2}A=1+{tan}^{2}A}$

Transposing the square from LHS to RHS.

$\twoheadrightarrow\quad\sf{sec\: A=\sqrt{1+{tan}^{2}A}}$

We know, that tan A equals to the reciprocal of cot A.

$\twoheadrightarrow\quad\sf{sec\: A=\sqrt{1+{\left(\dfrac{1}{cot}\right)}^{2}}}$

After futher solving,

$\twoheadrightarrow\quad\sf{sec\: A=\sqrt{1+\left(\dfrac{1}{{cot}^{2}A}\right)}}$

Performing LCM inside the square root.

$\twoheadrightarrow\quad\sf{sec\: A=\sqrt{\dfrac{{cot}^{2}A+1}{{cot}^{2}A}}}$

After futher solving.

$\twoheadrightarrow\quad\bold{sec\: A=\dfrac{\sqrt{{cot}^{2}A+1}}{cot\: A}}$

Therefore value of sec A in terms of cot A equals to √(cot²A+1)/cotA.

Now, let us calculate the value of tan A in terms of cot A. we know, tan A equals to the reciprocal of cot A.

$\twoheadrightarrow\quad\bold{tan\: A=\dfrac{1}{cot\: A}}$

Therefore, value of tan A in terms of cot A equals to 1/cot A.