To find the ratio of the times of flight for two projectiles fired from the ground with the same initial speed but at different angles, we follow these steps:

Projectile Angles:
The first projectile is launched at an angle .

The second projectile is launched at an angle .

Time of Flight Formula:

The time of flight for a projectile is given by:

where is the initial speed and is the

acceleration due to gravity.
Ratio of Times of Flight:
The ratio of the times of flights can be determined as follows:

Applying Trigonometric Identities:
Use the sine addition and subtraction formulas:

Simplifying the Ratio:
Substitute the expressions for and into the ratio:

Factor out the trigonometric functions to express in terms of tangent:

Therefore, the ratio of the times of flights for the two projectiles is .