The question asks to find an angle $\theta$ such that $0^\circ < \theta < 360^\circ$, which has:

1. The same sine value as $240^\circ$.
2. The same cosine value as $240^\circ$.

1. Finding $\theta$ with the same sine value as $240^\circ$

The sine function is periodic, and for a given angle, another angle can share the same sine value within one full revolution. The reference angle for $240^\circ$ is $60^\circ$ (since $240^\circ - 180^\circ = 60^\circ$).

Since sine is negative in the third and fourth quadrants, we need to find the angle in the fourth quadrant that has the same sine value as $240^\circ$.

The sine of $240^\circ$ is negative, and another angle that has the same sine value is $360^\circ - 240^\circ = 120^\circ$.

Thus, the angle with the same sine value as $240^\circ$ is $120^\circ$.

2. Cosine value of $240^\circ$

The cosine of $240^\circ$ is negative as well, and the angle that has the same cosine value is $120^\circ$, which is already given.

Would you like further explanation on trigonometric identities, or do you have any specific questions?

Related Questions:

1. How do we find reference angles for trigonometric functions?
2. What is the general solution for angles with the same sine value in trigonometry?
3. Can an angle have the same sine and cosine values?
4. Why is the sine of angles in the third and fourth quadrants negative?
5. How do we use the unit circle to find angles with the same sine or cosine values?

Tip:

Always use the unit circle to check the signs of sine and cosine for angles in different quadrants—it helps in finding equivalent angles!