Question

what is the value of cosec 75 degree​

Answer 1

Proof

csc75∘ = 1sin75∘ Cosecant is Reciprocal of Sine

= 16√+2√4 Sine of 75∘

= 46–√+2–√ multiplying top and bottom by 4

= 4(6–√−2–√)(6–√+2–√)(6–√−2–√) multiplying top and bottom by 6–√−2–√

= 4(6–√−2–√)6−2 Difference of Two Squares

= 6–√−2–√ simplifying

Answer 2

Value of cosec75° is 1.035.

Given:

• $cosec \: {75}^{ \circ}$

To find:

• Find the value of $cosec \: {75}^{ \circ}$

Solution:

Formula to be used:

1. $cosec \: \theta = \frac{1}{sin \: \theta}$
2. $\sin(A+B) = \sin(A) \cos(B) + \cos(A) \sin(B)$

Step 1:

Apply trigonometric identity 1.

$cosec \: {75}^{ \circ} = \frac{1}{sin \: {75}^{ \circ}}...eq1$

Step 2:

Find the value of sin 75° using formula 2.

As,

75° can be written as 30°+45°.

So,

$\sin( {75}^{ \circ}) = \sin({30}^{ \circ} + {45}^{ \circ})$

or

$\sin({30}^{ \circ} + {45}^{ \circ}) = \sin({30}^{ \circ}) \cos({45}^{ \circ}) + \cos({30}^{ \circ}) \sin({45}^{ \circ})$

Put the values of angles.

As

$\sin({30}^{ \circ})=\frac{1}{2}$

$\cos({45}^{ \circ})= \frac{1}{\sqrt{2}}$

$\cos({30}^{ \circ}) =\frac{\sqrt{3}}{2}$

$sin({45}^{ \circ})= \frac{1}{\sqrt{2}}$

So,

$\sin({30}^{ \circ} + {45}^{ \circ}) = \frac{1}{2} \times \frac{1}{ \sqrt{2} } + \frac{ \sqrt{3} }{2} \times \frac{1}{ \sqrt{2} }$

or

$\sin({30}^{ \circ} + {45}^{ \circ}) = \frac{1}{2 \sqrt{2} } + \frac{ \sqrt{3} }{2 \sqrt{2} }$

or

$\bf \sin({75}^{ \circ}) = \frac{1 + \sqrt{3} }{2 \sqrt{2} }$

Step 3:

Put the value of sin75° in eq1.

$cosec \: {75}^{ \circ} = \frac{1}{ \frac{1 + \sqrt{3} }{2 \sqrt{2} } }$

or

$cosec \: {75}^{ \circ} = \frac{2 \sqrt{2} }{ 1 + \sqrt{3} }$

Step 4:

Put the value of √2 and √3.

As √2= 1.414

√3= 1.732

$cosec \: {75}^{ \circ} = \frac{2 \times 1.414 }{ 1 + 1.732 }$

or

$cosec \: {75}^{ \circ} = \frac{2.82 8 }{ 2.732 }$

or

$\bf cosec \: {75}^{ \circ} = 1.035$

Thus,

Value of cosec75° is 1.035.

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