Question

find an angle $\theta$ with $0^{circ}<\theta<360^{circ}$ that has the same:
sine as $200^{circ}$: $\theta = square$ degrees
cosine as $200^{circ}$: $\theta = 160$ degrees
enter an integer or decimal number (more...)

Explanation:

Step1: Recall sine - angle property

The sine function has the property $\sin\theta=\sin(180^{\circ}+\alpha)$ for $\theta = 180^{\circ}+\alpha$ and also $\sin\theta=\sin(180^{\circ}-\alpha)$. Given $\alpha = 20^{\circ}$ (since $200^{\circ}=180^{\circ} + 20^{\circ}$), then $\sin200^{\circ}=\sin(180^{\circ}+ 20^{\circ})$. Another angle with the same sine value in the range $0^{\circ}<\theta<360^{\circ}$ is $\theta = 180^{\circ}-20^{\circ}=160^{\circ}$.

Step2: Recall cosine - angle property

The cosine function has the property $\cos\theta=\cos(360^{\circ}-\alpha)$. Given $\alpha = 200^{\circ}$, then $\cos200^{\circ}=\cos(360^{\circ} - 200^{\circ})=\cos160^{\circ}$.

Answer:

Sine as $200^{\circ}$: $160$
Cosine as $200^{\circ}$: $160$