Selena Cowan

Answered

2022-01-23

what is x for ${\mathrm{tan}}^{2}3x=2{\mathrm{sin}}^{2}3x$

Answer & Explanation

trasahed

Expert

2022-01-24Added 14 answers

So we have either

or

From the above, the only admissible solutions are

The sum of these angles is

Allison Compton

Expert

2022-01-25Added 16 answers

Do some trigonometry and find the general solution first:

${\mathrm{tan}}^{2}3x=2{\mathrm{sin}}^{2}3x=\frac{2{\mathrm{tan}}^{2}3x}{1+{\mathrm{tan}}^{2}3x}\iff {\mathrm{tan}}^{2}3x(1+{\mathrm{tan}}^{2}3x)=2{\mathrm{tan}}^{2}3x$

So,

either$\mathrm{tan}3x=0\iff 3x\equiv 0\text{}\text{mod}\text{}180\iff x\equiv 0\text{}\text{mod}\text{}60$ ;

or${\mathrm{tan}}^{2}3x=1\iff \mathrm{tan}3x=\pm 1\iff 3x\equiv \pm 45\text{}\text{mod}\text{}180\iff x\equiv \pm 15\text{}\text{mod}\text{}60$

Then select the values in [0,90]

So,

either

or

Then select the values in [0,90]

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