Question

what is the smallest angle of this triangle? 17 16 29 ?° round to the nearest hundredth.

Explanation:

Step1: Identify target angle

The smallest angle is opposite the shortest side (length 16).

Step2: Apply Law of Cosines

Let $a=16$, $b=17$, $c=29$, angle $\theta$ opposite $a$.
$$\cos\theta = \frac{b^2 + c^2 - a^2}{2bc}$$

Step3: Substitute values

$$\cos\theta = \frac{17^2 + 29^2 - 16^2}{2(17)(29)}$$
$$\cos\theta = \frac{289 + 841 - 256}{986}$$
$$\cos\theta = \frac{874}{986} \approx 0.8864$$

Step4: Calculate inverse cosine

$$\theta = \cos^{-1}(0.8864) \approx 27.66^\circ$$

Answer:

$27.66^\circ$