If $$\cos3\theta=\sin(\theta-34^{\circ})$$, then the value of $$\theta$$ as an acute angle is:

Given,  $$\cos3\theta=\sin(\theta-34^{\circ})$$

$$\Rightarrow$$  $$\cos3\theta=\cos\left[90-(\theta-34^{\circ})\right]$$

$$\Rightarrow$$  $$\cos3\theta=\cos\left[90-\theta+34^{\circ}\right]$$

$$\Rightarrow$$  $$\cos3\theta=\cos\left[124^{^{\circ}}-\theta\right]$$

$$\Rightarrow$$  $$3\theta=124^{^{\circ}}-\theta$$

$$\Rightarrow$$  $$4\theta=124^{^{\circ}}$$

$$\Rightarrow$$  $$\theta=31^{^{\circ}}$$

Hence, the correct answer is Option C