Question

If cot theta equal to 1 by root 3 then find the value of sin 3 theta​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{cot\,\theta=\dfrac{1}{\sqrt{3}}}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}\;\mathsf{sin\,3\theta}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{cot\,\theta=\dfrac{1}{\sqrt{3}}}$

$\textsf{Taking reciprocals on bothsides, we get}$

$\mathsf{tan\,\theta=\sqrt{3}}$

$\implies\mathsf{\theta=\dfrac{\pi}{3}}$

$\mathsf{Now,}$

$\mathsf{sin\,3\theta}$

$\mathsf{=sin\,3\left(\dfrac{\pi}{3}\right)}$

$\mathsf{=sin\,\pi}$

$\mathsf{=0}\;\;\;\;\mathsf{(\because\;sin\,180^\circ=0)}$

$\implies\boxed{\mathsf{sin\,3\theta=0}}$

$\textbf{Standard trigonometric table:}$

$\left\begin{array}{|c|c|c|c|c|c|}\cline{1-6}&0^{\circ}&30^{\circ}&45^{\circ}&60^{\circ}&90^{\circ}\\\cline{1-6}\bf\,sin\theta&0&\frac{1}{2}&\frac{1}{\sqrt2}&\frac{\sqrt3}{2}&1\\\cline{1-6}\bf\,cos\theta&1&\frac{\sqrt3}{2}&\frac{1}{\sqrt2}&\frac{1}{2}&0\\\cline{1-6}\bf\,tan\theta&0&\frac{1}{\sqrt3}&1&\sqrt3&\infty\\\cline{1-6}\end{array}\right$

Answer 2

Answer:

sin3∅=0

Step-by-step explanation:

cot∅ = $1/\sqrt{3}$

⇒ tan∅ = √3

Trigonometry Values:

Angle 0 30          45        60         90

Sin θ 0 1/2           1/√2    √3/2    1

Cos θ 1 √3/2 1/√2 1/2         0

Tan θ 0 1/√3    1        √3              ∞

Cot θ ∞ √3         1         1/√3          0

Sec θ 1 2/√3 √2            2            ∞

Cosec θ  ∞ 2         √2    2/√3 1

∅=60 degrees or ∅=π/3

∴sin3∅ = sin3(60)

   sin3∅ = sin(180)  (sin180 belongs to second quadrants sine will be      positive)

    sin3∅ =sin(180 + 0)

     sin3∅=sin(0)  (the value of sin 0 = 0)

∴   sin3∅ = 0