Question

How do you find the exact functional value $\cos (\dfrac{{7\pi }}{{12}})$ using the cosine sum or difference identity?

Answer 1

Apply the formula of cosine sum $\cos (A + B) = \cos A\cos B - \sin A\sin B$.
In these questions try to break $\dfrac{{7\pi }}{{12}}$ in two terms so that you can use the above formula and denote A and B.

Complete step by step answer:
Firstly let's break $\dfrac{{7\pi }}{{12}}$ in two terms . We know $4 + 3 = 7$ , so we will break $\dfrac{{7\pi }}{{12}}$ in 4 and 3 . We cannot break it in other numbers because 12 is divisible by 4 and 3 and it is easier to calculate.
$\cos (\dfrac{{7\pi }}{{12}}) = \cos (\dfrac{{4\pi }}{{12}} + \dfrac{{3\pi }}{{12}})$
Simplifying the above
$\Rightarrow \cos (\dfrac{\pi }{3} + \dfrac{\pi }{4})$
Now , let’s take A as $\dfrac{\pi }{4}$ and B as $\dfrac{\pi }{3}$
Put the values in the formula $\cos (A + B) = \cos A\cos B - \sin A\sin B$
Calculating cos and sin separately to avoid confusion
For cosA and cosB
$\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and $\cos \dfrac{\pi }{3} = \dfrac{1}{2}$
For sinA and sinB
$\sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ and $\sin \dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}$
$\Rightarrow \cos (\dfrac{\pi }{3} + \dfrac{\pi }{4}) = \cos \dfrac{\pi }{4}\cos \dfrac{\pi }{3} - \sin \dfrac{\pi }{4}\sin \dfrac{\pi }{3}$
Putting the values of cosine and sine
$\Rightarrow \dfrac{1}{{\sqrt 2 }} \cdot \dfrac{1}{2} - \dfrac{1}{{\sqrt 2 }}\dfrac{{\sqrt 3 }}{2}$
Multiplying the above
$\Rightarrow \dfrac{1}{{2\sqrt 2 }} - \dfrac{{\sqrt 3 }}{{2\sqrt 2 }}$
Bothe have same denominators , subtract them
$\Rightarrow \dfrac{{1 - \sqrt 3 }}{{2\sqrt 2 }}$

Thus , value of $\cos (\dfrac{{7\pi }}{{12}})$ is $\dfrac{{1 - \sqrt 3 }}{{2\sqrt 2 }}$.

Additional information:
You can check from the calculator if the answer we obtained is correct or not.
For $\cos (\dfrac{{7\pi }}{{12}})$ , finding the value using calculator we get -0.259
For $\dfrac{{1 - \sqrt 3 }}{{2\sqrt 2 }}$ , finding the value using calculator we get -0.259
Therefore our obtained value is correct.

Note:
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values.
Therefore in the above question, we have used the cosine sum formula, if you have tried to break it into different identity it would be difficult to know the complex angles and it will take a long time.
For example when you break $\cos (\dfrac{{7\pi }}{{12}})$ for difference identity , you will break 7 in 10 and 3 then you will get angles in $\dfrac{{10\pi }}{{12}}$ or $\dfrac{{5\pi }}{{12}}$ and $\dfrac{\pi }{4}$ . Here finding the value of $\dfrac{{5\pi }}{{12}}$ will be a long calculation.
So use the formulas accordingly.