Answer

Verified

338.4k+ views

**Hint:**Use the half angle formula and write down an expression of for sin(67.5) in terms of sine of double of this angle, which is 135 degrees. Then find the value of cosine of 135 degrees and substitute in the half angle formula.

**Complete step by step solution:**

Let us first simplify the given trigonometric ratio, i.e. sin(67.5). Here, let us assume that the angle 67.5 is in degrees. In the given question it is asked us to calculate the value of sin(67.5).

The half angle formula says that

${{\sin }^{2}}\left( \dfrac{x}{2} \right)=\dfrac{1-\cos x}{2}$ …. (1),

where $\dfrac{x}{2}$ is the half angle and x is the double of that angle.

In this case, the half angle is equal to 67.5 degrees. This means that $\dfrac{x}{2}={{67.5}^{\circ }}$

Which further means that $x=2\times {{67.5}^{\circ }}={{135}^{\circ }}$

Substitute the values of the angles x and half of x in equation (1).

This gives us that ${{\sin }^{2}}\left( {{67.5}^{\circ }} \right)=\dfrac{1-\cos ({{135}^{\circ }})}{2}$ …. (i)

Now, let us find the value of $\cos ({{135}^{\circ }})$.

We can write 135 as $180-45$.

Therefore, $\cos ({{135}^{\circ }})$ can be written as $\cos ({{135}^{\circ }})=\cos ({{180}^{\circ }}-{{45}^{\circ }})$.

But we know that $\cos ({{180}^{\circ }}-\theta )=-\cos (\theta )$ …. (ii)

In this case, $\theta ={{45}^{\circ }}$.

Therefore, substitute $\theta ={{45}^{\circ }}$ in equation (ii).

Then, we get that $\cos ({{180}^{\circ }}-{{45}^{\circ }})=-\cos ({{45}^{\circ }})$ ….. (iii).

We know that $\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}$

After substituting this value in equation (iii) we get that $\cos ({{180}^{\circ }}-{{45}^{\circ }})=-\dfrac{1}{\sqrt{2}}$.

This means that $\cos ({{135}^{\circ }})=\cos ({{180}^{\circ }}-{{45}^{\circ }})=-\dfrac{1}{\sqrt{2}}$.

Now, substitute this value in equation (i).

This gives us that ${{\sin }^{2}}\left( {{67.5}^{\circ }} \right)=\dfrac{1-\left( -\dfrac{1}{\sqrt{2}} \right)}{2}=\dfrac{1+\dfrac{1}{\sqrt{2}}}{2}$

$\Rightarrow {{\sin }^{2}}\left( {{67.5}^{\circ }} \right)=\dfrac{1+\sqrt{2}}{2\sqrt{2}}$

Now, take square roots on both the sides.

Then, this means that $\sin \left( {{67.5}^{\circ }} \right)=\pm \sqrt{\dfrac{1+\sqrt{2}}{2\sqrt{2}}}$.

But here 67.5 degrees is an acute angle and we know that sine of an acute angle is a positive number. Therefore, we discard the negative value, i.e. $\sin \left( {{67.5}^{\circ }} \right)=-\sqrt{\dfrac{1+\sqrt{2}}{2\sqrt{2}}}$ and consider $\sin \left( {{67.5}^{\circ }} \right)=\sqrt{\dfrac{1+\sqrt{2}}{2\sqrt{2}}}$.

**Hence, we found with the help of half angle formula that $\sin \left( {{67.5}^{\circ }} \right)=\sqrt{\dfrac{1+\sqrt{2}}{2\sqrt{2}}}$.**

**Note:**Sometimes in some questions, the formulae may help to solve the questions.

$\sin (2\pi +\theta )=\sin (\theta )$

$\Rightarrow\cos (2\pi +\theta )=\cos (\theta )$

$\Rightarrow\sin (\pi +\theta )=-\sin (\theta )$

$\Rightarrow\cos (\pi +\theta )=-\cos (\theta )$

$\Rightarrow\sin (-\theta )=-\sin (\theta )$

$\Rightarrow\cos (-\theta )=\cos (\theta )$

With the help of these formulae you can find the formulae for the other trigonometric ratios as all the other trigonometric ratios depend on sine and cosine.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a book review which you have recently read in class 8 english CBSE

Difference Between Plant Cell and Animal Cell

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

State the laws of reflection of light

What I want should not be confused with total inactivity class 12 english CBSE