Answer
Verified
386.9k+ views
Hint: Form the table of trigonometric values with angles such as \[{{0}^{\circ }},{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }},{{90}^{\circ }}\]and with trigonometric function sine, cosine, tangent. Find the value of trigonometric functions from the question and substitute to find the required answer.
Complete step-by-step answer:
Given, \[\tan 3x=\sin 45.\cos 45+\sin 30\].
We can find the values of RHS using basic trigonometric formulas.
We can find it by creating a trigonometric table with the required angles such as \[{{0}^{\circ }},{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }}\]and \[{{90}^{\circ }}\]. With 6 trigonometric functions such as sine, cosine, tangent, cosecant, secant and cotangent.
We can draw the table.
Now considering the RHS = \[\sin 45.\cos 45+\sin 30\].
From the above table we can find the values of sin45, cos45 and sin30.
Value of \[\sin 45=\dfrac{1}{\sqrt{2}}\].
Value of \[\cos 45=\dfrac{1}{\sqrt{2}}\].
Value of \[\sin 30=\dfrac{1}{2}\].
Substituting the values in RHS we get,
\[\begin{align}
& RHS=\sin 45.\cos 45+\sin 30 \\
& =\dfrac{1}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}}+\dfrac{1}{2}=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{2}{2}=1 \\
\end{align}\]
Given, \[LHS=\tan 3x\], putting the value of RHS, we get,
\[\begin{align}
& \tan 3x=1 \\
& \Rightarrow 3x{{\tan }^{-1}}\left( 1 \right) \\
\end{align}\]
From the table, \[\tan 45=1\].
Taking the \[{{\tan }^{-1}}\left( 1 \right)\]we get \[{{45}^{\circ }}\].
\[\begin{align}
& \therefore 3x=45 \\
& \Rightarrow x=\dfrac{45}{3}={{15}^{\circ }} \\
\end{align}\]
Therefore, we get the values of x as \[{{15}^{\circ }}\].
Note:
We got \[{{\tan }^{-1}}\left( 1 \right)={{45}^{\circ }}\].
We know \[{{\tan }^{-1}}\theta =\cot \theta \]i.e. \[\dfrac{1}{\tan \theta }=\cot \theta \].
From the table of the trigonometric functions. Or try to remember the first 3 functions \[\sin \theta ,\cos \theta \] and \[\tan \theta \]. The other 3 functions can be found taking the reverse of \[\sin \theta \] (to get \[\cos ec\theta \]), \[\cos \theta \] (to get \[\sec \theta \]) and \[\tan \theta \] (to get \[\cot \theta \]).
Complete step-by-step answer:
Given, \[\tan 3x=\sin 45.\cos 45+\sin 30\].
We can find the values of RHS using basic trigonometric formulas.
We can find it by creating a trigonometric table with the required angles such as \[{{0}^{\circ }},{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }}\]and \[{{90}^{\circ }}\]. With 6 trigonometric functions such as sine, cosine, tangent, cosecant, secant and cotangent.
We can draw the table.
Now considering the RHS = \[\sin 45.\cos 45+\sin 30\].
From the above table we can find the values of sin45, cos45 and sin30.
Value of \[\sin 45=\dfrac{1}{\sqrt{2}}\].
Value of \[\cos 45=\dfrac{1}{\sqrt{2}}\].
Value of \[\sin 30=\dfrac{1}{2}\].
Substituting the values in RHS we get,
\[\begin{align}
& RHS=\sin 45.\cos 45+\sin 30 \\
& =\dfrac{1}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}}+\dfrac{1}{2}=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{2}{2}=1 \\
\end{align}\]
Given, \[LHS=\tan 3x\], putting the value of RHS, we get,
\[\begin{align}
& \tan 3x=1 \\
& \Rightarrow 3x{{\tan }^{-1}}\left( 1 \right) \\
\end{align}\]
From the table, \[\tan 45=1\].
Taking the \[{{\tan }^{-1}}\left( 1 \right)\]we get \[{{45}^{\circ }}\].
\[\begin{align}
& \therefore 3x=45 \\
& \Rightarrow x=\dfrac{45}{3}={{15}^{\circ }} \\
\end{align}\]
Therefore, we get the values of x as \[{{15}^{\circ }}\].
Note:
We got \[{{\tan }^{-1}}\left( 1 \right)={{45}^{\circ }}\].
We know \[{{\tan }^{-1}}\theta =\cot \theta \]i.e. \[\dfrac{1}{\tan \theta }=\cot \theta \].
From the table of the trigonometric functions. Or try to remember the first 3 functions \[\sin \theta ,\cos \theta \] and \[\tan \theta \]. The other 3 functions can be found taking the reverse of \[\sin \theta \] (to get \[\cos ec\theta \]), \[\cos \theta \] (to get \[\sec \theta \]) and \[\tan \theta \] (to get \[\cot \theta \]).
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE