### Trigonometry: Trigonometric functions

### Trigonometric equations 2

We have seen how we can solve trigonometric equations of the form #\sin(ax+b)=c#, #\cos(ax+b)=c# and #\tan(ax+b)=c#. There is still another form of trigonometric equations we can easily solve, namely #\sin(A)=\sin(B)#, #\cos(A)=\cos(B)# and #\tan(A)=\tan(B)#, in which #A# and #B# are expressions in #x#.

The solution of the equation #\sin(\blue A)=\sin(\green B)#, in which #\blue A# and #\green B# are expressions in #x# has the following solutions:

\[\blue A=\green B+k \cdot 2 \pi \lor \blue A=\pi-\green B+k\cdot 2\pi\]

Here, #k# is an integer.

**Example**

\[\sin(\blue{x+\pi})=\sin(\green{2x})\]

has solutions

\[\blue{x+\pi}=\green{2x}+k\cdot2 \pi \lor \blue{x+\pi}=\pi-\green{2x}+k \cdot 2 \pi\]

The solution of the equation #\cos(\blue A)=\cos(\green B)#, in which #\blue A# and #\green B# are expressions in #x# has the following solutions:

\[\blue A=\green B+k \cdot 2 \pi \lor \blue A=-\green B+k\cdot 2\pi\]

Here, #k# is an integer.

**Example**

\[\cos(\blue{x+\pi})=\cos(\green{2x})\]

has solutions

\[\blue{x+\pi}=\green{2x}+k\cdot2 \pi \lor \blue{x+\pi}=-\green{2x}+k \cdot 2 \pi\]

The solution of the equation #\tan(\blue A)=\tan(\green B)#, in which #\blue A# and #\green B# are expressions in #x#, has the following solutions:

\[\blue A=\green B+k \cdot \pi \]

Here, #k# is an integer.

**Example**

\[\tan(\blue{x+\pi})=\tan(\green{2x})\]

has solutions

\[\blue{x+\pi}=\green{2x}+k\cdot \pi\]

Give your answer in the form: #x=x_1 \lor x=x_2 \lor \ldots x=x_n#, in which #x_1#, #x_2#, #\ldots#, #x_n# are the correct solutions and #n# is the number of solutions. Use #k# as a random integer.

#\begin{array}{rcl}

\sin\left({{\pi}\over{6}}+2\cdot x\right)&=&\sin\left({{\pi}\over{6}}+4\cdot x\right) \\ &&\phantom{xxx}\blue{\text{the equation we need to solve}}\\

{{\pi}\over{6}}+2\cdot x={{\pi}\over{6}}+4\cdot x+2 \pi \cdot k &\lor& {{\pi}\over{6}}+2\cdot x=\pi-({{\pi}\over{6}}+4\cdot x)+2 \pi \cdot k \\ &&\phantom{xxx}\blue{\text{general rule for solution }\sin(A)=\sin(B)}\\

x=\left(-\pi\right)\cdot k &\lor& x={{\pi}\over{9}}+{{\pi\cdot k}\over{3}}\\ &&\phantom{xxx}\blue{\text{reduced}}\\

\end{array}#

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