Let's solve the problem step by step:

1. Find an angle with the same sine as $50^\circ$:

   • The sine function is positive in the first and second quadrants.
   • The angle in the second quadrant that has the same sine as $50^\circ$ is given by: $\theta = 180^\circ - 50^\circ = 130^\circ$ So, $\theta = 130^\circ$.

2. Find an angle with the same cosine as $50^\circ$:

   • The cosine function is positive in the first and fourth quadrants.
   • The angle in the fourth quadrant that has the same cosine as $50^\circ$ is given by: $\theta = 360^\circ - 50^\circ = 310^\circ$ So, $\theta = 310^\circ$.

Final Answers:

• For the same sine as $50^\circ$: $\theta = 130^\circ$.
• For the same cosine as $50^\circ$: $\theta = 310^\circ$.

Do you want me to explain this further or solve similar problems?

Related Questions:

1. What are the general solutions for angles with the same sine or cosine in a full rotation?
2. How does the unit circle help in determining angles with the same sine or cosine?
3. What are the properties of sine and cosine functions in each quadrant?
4. How are sine and cosine related to complementary angles?
5. Can we determine these angles using radians instead of degrees?

Tip: Always remember the reference angle rules in trigonometry for determining equivalent angles in different quadrants!