(d) $$ \sin heta=1 $$ (g) $$ \sin heta-0,574=0 $$

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(d) $$ \sin 	heta=1 $$ (g) $$ \sin 	heta-0,574=0 $$

UpStudy AI · Accepted Answer

**(d)** Given the equation
$$
\sin \theta = 1,
$$
we know that the sine function reaches its maximum value $$1$$ only at an angle of $$\frac{\pi}{2}$$ within the interval $$[0, 2\pi)$$. Because sine has a period of $$2\pi$$, the general solution is
$$
\theta = \frac{\pi}{2} + 2\pi k,\quad \text{where } k \in \mathbb{Z}.
$$

**(g)** The equation
$$
\sin \theta - 0.574 = 0
$$
can be rewritten as
$$
\sin \theta = 0.574.
$$
Let
$$
\alpha = \arcsin(0.574).
$$
Within a period, the sine function takes the value $$0.574$$ at two angles:
$$
\theta = \alpha \quad \text{and} \quad \theta = \pi - \alpha.
$$
Taking into account the periodicity of the sine function (period $$2\pi$$), the general solutions are:
$$
\theta = \alpha + 2\pi k \quad \text{or} \quad \theta = (\pi - \alpha) + 2\pi k,\quad \text{where } k \in \mathbb{Z}.
$$
**(d)** The angle $$\theta$$ is $$\frac{\pi}{2}$$ plus any multiple of $$2\pi$$. **(g)** The angles $$\theta$$ are $$\arcsin(0.574)$$ and $$\pi - \arcsin(0.574)$$, plus any multiple of $$2\pi$$.

(d) \( \sin \theta=1 \) (g) \( \sin \theta-0,574=0 \)

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