sin2x =12/13
cos2x = 5/13
tan2x = 12/5
STEP - BY - STEP EXPLANATION
What to find?
• sin2x
,
• cos2x
,
• tan2x
Given:
tanx = 2/3 = opposite / adjacent
We need to first make a sketch of the given problem.
Let h be the hypotenuse.
We need to find sinx and cos x, but to find sinx and cosx, first determine the value of h.
Using the Pythagoras theorem;
hypotenuse² = opposite² + adjacent²
h² = 2² + 3²
h² = 4 + 9
h² =13
Take the square root of both-side of the equation.
h =√13
This implies that hypotenuse = √13
We can now proceed to find the values of ainx and cosx.
Using the trigonometric ratio;
[tex]\sin x=\frac{opposite}{\text{hypotenuse}}=\frac{2}{\sqrt[]{13}}[/tex][tex]\cos x=\frac{adjacent}{\text{hypotenuse}}=\frac{3}{\sqrt[]{13}}[/tex]
And we know that tanx =2/3
From the trigonometric identity;
sin 2x = 2sinxcosx
Substitute the value of sinx , cosx and then simplify.
[tex]\sin 2x=2(\frac{2}{\sqrt[]{13}})(\frac{3}{\sqrt[]{13}})[/tex][tex]=\frac{12}{13}[/tex]
Hence, sin2x = 12/13
cos2x = cos²x - sin²x
Substitute the value of cosx, sinx and simplify.
[tex]\begin{gathered} \cos 2x=(\frac{3}{\sqrt[]{13}})^2-(\frac{2}{\sqrt[]{13}})^2 \\ \\ =\frac{9}{13}-\frac{4}{13} \\ =\frac{5}{13} \end{gathered}[/tex]
Hence, cos2x = 5/13
tan2x = 2tanx / 1- tan²x
[tex]\tan 2x=\frac{2\tan x}{1-\tan ^2x}[/tex][tex]=\frac{2(\frac{2}{3})}{1-(\frac{2}{3})^2}[/tex][tex]=\frac{\frac{4}{3}}{1-\frac{4}{9}}[/tex][tex]=\frac{\frac{4}{3}}{\frac{9-4}{9}}[/tex][tex]=\frac{\frac{4}{3}}{\frac{5}{9}}[/tex][tex]=\frac{4}{3}\times\frac{9}{5}=\frac{4}{1}\times\frac{3}{5}=\frac{12}{5}[/tex]
OR
[tex]\tan 2x=\frac{\sin 2x}{\cos 2x}=\frac{\frac{12}{13}}{\frac{5}{13}}=\frac{12}{5}[/tex]
Hence, tan2x = 12/5
Therefore,
sin2x =12/13
cos2x = 5/13
tan2x = 12/5
