Question

How do you find the exact functional value of \[\tan \left( {{45}^{0}}+{{30}^{0}} \right)\] using the cosine sum or difference identity?

Answer 1

Hint: In the given question, we have been asked to find the exact value of a given tan function using the cosine sum or difference trigonometric identity. Then solve the question by calculating as \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] , taking\[\theta =\left( {{45}^{0}}+{{30}^{0}} \right)\]. Then we continue by finding the sin and cos of the same angle as given and get the exact value of \[\tan \left( {{45}^{0}}+{{30}^{0}} \right)\].

Formula used:
To find the tangent of any angle we need to just divide the sine and cosine of the same angle:\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }=\dfrac{perpendicular}{base}\]

Trigonometric ratio table used to find the sine and cosine of the angle:

Angles(in degrees)	\[\sin \theta \]	\[\cos \theta \]
\[{{0}^{0}}\]	0	1
\[{{30}^{0}}\]	\[\dfrac{1}{2}\]	\[\dfrac{\sqrt{3}}{2}\]
\[{{45}^{0}}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{1}{\sqrt{2}}\]
\[{{60}^{0}}\]	\[\dfrac{\sqrt{3}}{2}\]	\[\dfrac{1}{2}\]
\[{{90}^{0}}\]	1	0

Complete step by step solution:
We have given that,
\[\Rightarrow \tan \left( {{45}^{0}}+{{30}^{0}} \right)\]
We know that:
\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], thus
\[\Rightarrow \tan \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\sin \left( {{45}^{0}}+{{30}^{0}} \right)}{\cos \left( {{45}^{0}}+{{30}^{0}} \right)}\]
Now solving,
\[\Rightarrow \sin \left( {{45}^{0}}+{{30}^{0}} \right)\]
Using the identity; \[\sin \left( a+b \right)=\sin a\cos b+\cos a\sin b\], we obtain
\[\Rightarrow \sin \left( {{45}^{0}}+{{30}^{0}} \right)=\sin {{45}^{0}}\cos {{30}^{0}}-\cos {{45}^{0}}\sin {{30}^{0}}\]
Using the trigonometric ratios table;
\[\Rightarrow \sin \left( {{45}^{0}}+{{30}^{0}} \right)=\sin {{45}^{0}}\cos {{30}^{0}}-\cos {{45}^{0}}\sin {{30}^{0}}=\left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{\sqrt{2}}{2} \right)+\left( \dfrac{1}{2} \right)\left( \dfrac{\sqrt{2}}{2} \right)\]
Simplifying the numbers, we get
\[\Rightarrow \sin \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\sqrt{2}}{4}\left( \sqrt{3}+1 \right)\]
Now solving,
\[\Rightarrow \cos \left( {{45}^{0}}+{{30}^{0}} \right)\]
Using the identity; \[\cos \left( a+b \right)=\cos a\cos b+\sin a\sin b\], we obtain
\[\Rightarrow \cos \left( {{45}^{0}}+{{30}^{0}} \right)=\cos {{45}^{0}}\cos {{30}^{0}}-\sin {{45}^{0}}\sin {{30}^{0}}\]
Using the trigonometric ratios table;
\[\Rightarrow \cos \left( {{45}^{0}}+{{30}^{0}} \right)=\cos {{45}^{0}}\cos {{30}^{0}}-\sin {{45}^{0}}\sin {{30}^{0}}=\left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{\sqrt{2}}{2} \right)-\left( \dfrac{1}{2} \right)\left( \dfrac{\sqrt{2}}{2} \right)\]
Simplifying the numbers, we get
\[\Rightarrow \cos \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\sqrt{2}}{4}\left( \sqrt{3}-1 \right)\]
Thus,
We have
\[\Rightarrow \tan \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\sin \left( {{45}^{0}}+{{30}^{0}} \right)}{\cos \left( {{45}^{0}}+{{30}^{0}} \right)}\]
Putting the values from above, we get
\[\Rightarrow \tan \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\dfrac{\sqrt{2}}{4}\left( \sqrt{3}+1 \right)}{\dfrac{\sqrt{2}}{4}\left( \sqrt{3}-1 \right)}=\dfrac{\sqrt{3}+1}{\sqrt{3}-1}\]

Therefore,
\[\Rightarrow \tan \left( {{45}^{0}}+{{30}^{0}} \right)=\dfrac{\sqrt{3}+1}{\sqrt{3}-1}\]
Hence, it is the required answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.