Question

solve for x.
12 cm
x
70°
x = ? cm

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle. The side adjacent to the \(70^\circ\) angle is \(x\), and the side opposite is \(12\) cm. So we use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), but wait, actually, \(\tan(70^\circ)=\frac{12}{x}\) (since opposite to \(70^\circ\) is \(12\), adjacent is \(x\)). Wait, no: angle is \(70^\circ\), right angle, so the other angle is \(20^\circ\), but let's check. Wait, in the right triangle, one angle is \(90^\circ\), one is \(70^\circ\), so the sides: the side labeled \(12\) is opposite the \(70^\circ\) angle? Wait, no, the right angle is at the top left, so the sides: the vertical side is \(x\) (adjacent to \(70^\circ\)), the horizontal side is \(12\) (opposite to \(70^\circ\)). So \(\tan(70^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{12}{x}\). So we can solve for \(x\): \(x = \frac{12}{\tan(70^\circ)}\).

Step2: Calculate the value

We know that \(\tan(70^\circ)\approx 2.7475\). So \(x=\frac{12}{2.7475}\approx 4.37\) (rounded to two decimal places). Wait, or maybe I mixed up opposite and adjacent. Wait, let's re-examine the triangle. The right angle is at the top left, so the angle at the bottom left is \(70^\circ\). So the sides: the side from bottom left to top left is \(x\) (adjacent to \(70^\circ\)), the side from top left to top right is \(12\) (opposite to \(70^\circ\)). So yes, \(\tan(70^\circ)=\frac{12}{x}\), so \(x = \frac{12}{\tan(70^\circ)}\). Let's compute that. \(\tan(70^\circ)\approx 2.7475\), so \(12\div 2.7475\approx 4.37\). Alternatively, if we use \(\cot(70^\circ)=\frac{x}{12}\), since \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\), so \(x = 12\times\cot(70^\circ)\). And \(\cot(70^\circ)=\frac{1}{\tan(70^\circ)}\approx 0.3640\), so \(12\times 0.3640\approx 4.37\).

Answer:

\(x\approx 4.37\) (or more accurately, using calculator: \(\tan(70^\circ)\approx 2.747477419\), so \(12\div 2.747477419\approx 4.367\), so approximately \(4.37\) cm, or maybe rounded to one decimal place, \(4.4\) cm. But let's check with more precise calculation. Let's use \(\tan(70^\circ)=2.747477419\), so \(x = 12 / 2.747477419 \approx 4.367\), so \(\approx 4.37\) cm.