Question

i) triangle with hypotenuse 18 cm, angle 38° at the top, right angle at the bottom right, side x opposite the angle? wait, no, right angle at bottom right, so the side x is adjacent? wait, hypotenuse is 18, angle 38° at the top vertex. so the right angle is at the bottom right, so the sides: hypotenuse 18, angle 38° at the top, so the side x is adjacent to the 38° angle? wait, no, in a right triangle, the right angle is between the two legs. so if the right angle is at the bottom right, then the legs are the bottom (horizontal) and right (vertical, x). the hypotenuse is the left side, length 18, and the angle at the top (between hypotenuse and vertical leg x) is 38°. so to find x, we can use cosine: cos(38°) = adjacent/hypotenuse = x/18, so x = 18cos(38°). or maybe sine? wait, angle at the top: lets label the triangle: vertices a (top, 38°), b (bottom right, right angle), c (bottom left). so sides: ab is hypotenuse? no, ac is hypotenuse (18 cm), angle at a is 38°, right angle at b. so side ab is x (vertical leg), bc is horizontal leg, ac is hypotenuse (18). so angle at a: 38°, so cos(angle a) = adjacent/hypotenuse = ab/ac = x/18. so x = 18cos(38°). so the problem is to find x, given right triangle with hypotenuse 18, angle 38° at the top, find the adjacent side (x). so ocr text: i) triangle image 18 cm, 38°, x (vertical leg), right angle at b. then x ≈ ______

Explanation:

Step1: Identify trigonometric ratio

We have a right - triangle. The hypotenuse is \(18\) cm, and we want to find the adjacent side to the \(38^{\circ}\) angle. The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 38^{\circ}\), adjacent side is \(x\) and hypotenuse \(h = 18\) cm. So, \(\cos(38^{\circ})=\frac{x}{18}\).

Step2: Solve for \(x\)

To solve for \(x\), we can multiply both sides of the equation \(\cos(38^{\circ})=\frac{x}{18}\) by \(18\). So, \(x = 18\times\cos(38^{\circ})\).

We know that \(\cos(38^{\circ})\approx0.7880\) (using a calculator to find the cosine of \(38\) degrees). Then \(x=18\times0.7880 = 14.184\approx14.2\) (rounded to one decimal place) or \(14\) (rounded to the nearest whole number). If we use more precise value of \(\cos(38^{\circ})\approx0.788010754\), then \(x = 18\times0.788010754=14.18419357\approx14.2\) cm.

Answer:

\(x\approx14.2\) (or \(14\) if rounded to the nearest whole number)