Question

#Question 1
What is the value of the following?
$cosec( \frac{7\pi}{6} ) \: sec( \frac{5\pi}{3} )$
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Answer 1

Answer:

-4

Step-by-step explanation:

Given :

$\mathrm{cosec\bigg(\dfrac{7\pi}{6}\bigg)sec\bigg(\dfrac{5\pi}{3}\bigg)}$

To find :

the value of the following

Solution :

$\longmapsto \mathrm{cosec\bigg(\dfrac{7\pi}{6}\bigg)sec\bigg(\dfrac{5\pi}{3}\bigg)}$

$\implies \mathrm{cosec\bigg(\pi +\dfrac{\pi}{6}\bigg)sec\bigg(\pi + \dfrac{2\pi}{3}\bigg)}$

We know that,

$\boxed{\begin{array}{cc} \mathrm{cosec(\pi + x) = -cosecx}\\\mathrm{sec(\pi + x) = -secx}\end{array}}$

$\implies \mathrm{-cosec\bigg(\dfrac{\pi}{6}\bigg)\times -sec\bigg(\dfrac{2\pi}{3}\bigg)}$

$\implies \mathrm{cosec\bigg(\dfrac{\pi}{6}\bigg)sec\bigg(\dfrac{2\pi}{3}\bigg)}$

We know that,

$\boxed{\begin{array}{cc} \mathrm{cosec\bigg(\dfrac{\pi}{6}\bigg) = 2}\\\\\mathrm{sec\bigg(\dfrac{2\pi}{3}\bigg) = -2}\end{array}}$

So,

$\implies \mathrm{2(-2)}$

$\therefore \underline{\boxed{\mathbf{cosec\bigg(\dfrac{7\pi}{6}\bigg)sec\bigg(\dfrac{5\pi}{3}\bigg) = -4}}}$

❖ Extra information :

⟡ Trigonometric Full Table :

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

⟡ The CAST diagram :

$\setlength{\unitlength}{7mm}\begin{picture}(0,0)\thicklines\put(0,0){\vector(1,0){6}}\put(0,0){\vector(-1,0){6}}\put(0,0){\vector(0,1){6}}\put(0,0){\vector(0,-1){6}}\put( -3,4){\sf\huge S}\put(3,4){\sf\huge A}\put( -3, - 2){\sf\huge T}\put(3, - 2){\sf\huge C}\put( -4.5,2.5){\sf\large Sin is positive}\put(1.5,2.5){\sf\large All are positive}\put( -4.5, - 3){\sf\large Tan is positive}\put(2, - 3){\sf\large Cos is positive}\put( -4.1,1.5){\sf Cos and Tan }\put( -4.1,1){\sf are negative}\put( -4.1, -4){\sf Sin and Cos}\put( -4.1, - 4.5){\sf are negative}\put(2.4, - 4){\sf Sin and Tan}\put(2.4, - 4.5){\sf are negative}\end{picture}$

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Answer 2

$\mathbb\blue{ ANSWER}.$

cosec (750 +0) - sec(15⁰ - 0) - tan(55⁰ + 0) +

cot (35⁰ - 0)

= sec (90⁰ - 75⁰0) – sec(15⁰ - 0) -

cot (90° - 5500) + cot(35⁰ - 0)

= sec (15⁰0) – sec(15⁰ - 0) - cot(350 +

0) + cot (35⁰ + 0)

= 0