40-50-90 Triangle with Cosine Equation

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Triangle DEF contains right angle E. If angle D measures 40° and its adjacent side measures 7.6 units, what is the measure of side EF? Round your answer to the nearest hundredth.

So far, I have angle measures of 40, 50, and 90. Side DE is 7.6 units and the right ratio to use for this problem is cos, which is opp/adj.

The equation so far is c= E/cos40 degrees, the answer I got was c= E/0.766044443 but I have no clue where to go from there.

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From what I understood, we have the following where we have to find $x$:

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Yes, it’s a crude drawing, but you get the point.

Here, we would use $\tan(x)$, which is opposite over adjacent:

Here we have:

$\tan(40^\circ) = \frac{x}{7.6}$

Now to solve for $x$, we multiply both sides by $7.6$:

$x = 7.6 \cdot \tan(40)$

If you had a calculator, $\tan(40) \approx 0.84$.

Hence, $x = EF \approx 6.37 \text{ or } 6.38$.

In general, you can always just use the appropriate trig function and then solve for the unknown side $x$.