Question

8)
find the missing side. round to the nearest tenth.
31.1
31.4
31.2
31.3

Explanation:

Step1: Identify trigonometric ratio

The triangle is right-angled. The side labeled 14 is adjacent to the 66° angle, and \(x\) is the hypotenuse. Use cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Substitute values

$\cos(66^\circ) = \frac{14}{x}$

Step3: Solve for \(x\)

$x = \frac{14}{\cos(66^\circ)}$

Step4: Calculate value

$\cos(66^\circ) \approx 0.4067$, so $x \approx \frac{14}{0.4067} \approx 34.42$? Wait, no—wait, maybe the 14 is opposite? Wait, no, recheck: if 14 is opposite, then sine: $\sin(66^\circ)=\frac{14}{x} \Rightarrow x=\frac{14}{\sin(66^\circ)}\approx\frac{14}{0.9135}\approx15.33$ no. Wait, maybe 14 is adjacent, \(x\) is opposite: $\tan(66^\circ)=\frac{x}{14}\Rightarrow x=14\times\tan(66^\circ)\approx14\times2.2460\approx31.44$. Oh, maybe the angle is with the hypotenuse? Wait, the options include 31.1. Let's use calculator precision: $\tan(66^\circ)\approx2.24603677390421$. 14×2.24603677390421≈31.44451483465894. Rounded to nearest tenth is 31.4? But wait, maybe the angle is 65°? No, the problem says 66°. Wait, maybe I mixed adjacent and opposite. If 14 is opposite, then $\cot(66^\circ)=\frac{x}{14}\Rightarrow x=14\times\cot(66^\circ)\approx14\times0.4452\approx6.23$ no. Wait, the options have 31.1. Let's check $\cos(66^\circ)\approx0.40673664307580015$. 14/0.40673664307580015≈34.42. No. Wait, maybe the angle is 24°? No. Wait, maybe the triangle is labeled differently: the 14 is the hypotenuse, and \(x\) is adjacent? $\cos(66^\circ)=\frac{x}{14}\Rightarrow x=14\times0.4067≈5.69$ no. Wait, the options are around 31, so 14×2.2 is ~31. So $\tan(66^\circ)$ gives ~31.4, which is an option. But the first option is 31.1. Maybe the problem has a typo, but based on calculation, 14×tan(66°)≈31.4. So the answer is 31.4? Wait, no, let's use more precise calculation: tan(66) is approximately 2.2460367739. 14*2.2460367739 = 31.4445148346, which rounds to 31.4. So the answer is 31.4.

Wait, I think I made a mistake earlier. The correct calculation is 14 times tan(66 degrees) equals approximately 31.4, which is one of the options. So the answer is 31.4.

Step1: Choose trigonometric function

The side 14 is adjacent to 66°, \(x\) is opposite. Use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Substitute values

$\tan(66^\circ)=\frac{x}{14}$

Step3: Solve for \(x\)

$x=14\times\tan(66^\circ)$

Step4: Calculate and round

$\tan(66^\circ)\approx2.2460$, so $x\approx14\times2.2460=31.444\approx31.4$

Answer:

31.1