Given a right-angled triangle XYZ with angles Y=π/2 and Z=0.66, side y=2.1 mm find side z. Give your answer in mm to 2 decimal places.

Steps to Solve

1. Identify the remaining angle

In a triangle, the sum of the angles is $\pi$ (or 180 degrees). We can find angle X by subtracting the given angles from $\pi$: $$ X = \pi - \left(\frac{\pi}{2} + 0.66\right) $$

2. Calculate the exact value of angle X

We need to compute the value of angle X: $$ X = \pi - \frac{\pi}{2} - 0.66 $$ This simplifies to: $$ X = \frac{\pi}{2} - 0.66 $$

3. Use the sine relationship

Since we are working with a right triangle, we can use the sine function. The sine of an angle is equal to the length of the opposite side divided by the hypotenuse: $$ \sin(Z) = \frac{y}{z} $$ Substituting the known values ($y = 2.1$ mm and $Z = 0.66$ radians): $$ \sin(0.66) = \frac{2.1}{z} $$

4. Rearrange to solve for z

We need to rearrange the equation to isolate $z$: $$ z = \frac{2.1}{\sin(0.66)} $$

5. Calculate the value of z

Finally, we compute the length of side $z$: $$ z = \frac{2.1}{\sin(0.66)} $$