Basic trigonometry - Find length of side when knowing one side length and the opposite angle

I have this basic trigonometry question of finding the length of "x" in the triangle Finding X. I know one side of the triangle and the opposite angle, so I figured it should be a simple case of just filling in the equations of cosine. I've tried looking at other solutions of similar problems, but I can't wrap my head around how to apply their approach to my problem.

Sorry if this is way too basic, I notice it's been too long since I've used trig..

• Use the tangent (or cotangent) instead of the cosine, since you aren't given the hypotenuse. – saulspatz Mar 14 at 14:10
• Actually, according to your figure, you know two angles: the $6$-degree angle and the right angle. That is what makes this problem solvable. – David K Mar 14 at 14:12
• This may be helpful in applying trigonometry to right triangles: intmath.com/trigonometric-functions/… – David K Mar 14 at 14:17

$$\tan \theta =\dfrac{\text{side opposite }\theta}{\text{side adjacent }\theta}$$

Here the $$\text{side opposite }\theta$$ to the angle of $$6^{\circ}$$ is $$x$$ and $$\text{side adjacent }\theta$$ is $$3$$.

$$\tan 6^{\circ}=\dfrac{x}{3}\implies x=3\tan6^{\circ}$$

You can leave it like that as $$6^{\circ}$$ is not a special angle, if however, you can use a calculator, simply feed in the expression to get $$x \approx 0.315312705798$$ and you're done.

• Have a look at mathworld.wolfram.com/TrigonometryAnglesPi30.html – Claude Leibovici Mar 14 at 15:03
• Thanks for the reference @ClaudeLeibovici I meant special angle in the sense those that're usually remembered by most people like $\{0, 30, 45, 60, 90\}$ for the acute angle case. A value like $\tan 6^{\circ}$ is best to be left computed than remembered. – Paras Khosla Mar 14 at 15:09
• For sure, I agree ! But, it is nice to know that we have an exact result. Replace the $3$ meters by $12345678987654321$ km and we can produce the exact result say within an error of $10^{-10}$ microns ! Cheers :-) – Claude Leibovici Mar 14 at 15:14
• Sure that is true @ClaudeLeibovici. Cheers – Paras Khosla Mar 14 at 15:15
• Sure @ClaudeLeibovici "Numerical Efficiency is your main concern" :) – Paras Khosla Mar 15 at 6:17

Use tangent: $$\tan 6°=\frac{x}{3} \rightarrow x=3\tan 6° \approx 0.32$$