Javelin throw, find time it takes for angle one to change to angle two

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The discussion centers on calculating the time it takes for the angle of a javelin throw to change from 34° to 15° after being launched at a speed of 27 m/s. The participant initially uses the equation t = (Vf - Vi) / a but struggles with deriving a combined equation for the problem. They attempt to find the final velocity using trigonometric components but initially arrive at an incorrect answer of 0.390 seconds. Ultimately, the participant discovers the correct solution after further consideration. The conversation highlights the importance of understanding both the underlying physics concepts and the mathematical manipulations involved.
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1. Homework Statement
In the javelin throw at a track-and-field event, the javelin is launched at a speed of 27 m/s at an angle of 34° above the horizontal. As the javelin travels upward, its velocity points above the horizontal at an angle that decreases as time passes. How much time is required for the angle to be reduced from 34° at launch to 15°?

Vi=27m/s
ay (g) = -9.8m/s^2
ax = 0m/s^2

2. Homework Equations
Vaverage=(Vf+Vi)/2=d/t
a=(Vf-Vi)/t

My teacher wants us to derive all other equations from this...I understand the why part of physics, it's just the "how" that I have trouble with. My teacher calls it "mathematical massaging". Also, he combines all of his equations into one giant one so that he only has to input his numbers once, no need to press "enter" several times. If you could please demonstrate how to "massage" the given equations and combine them into the larger final equation, that would be a great help.

3. The Attempt at a Solution
t=(Vf-Vi)/a
t=((cos34 deg)*(27m/s)*(sec15deg)-27m/s)/-9.8m/s^2
The reason I multiply cos34*27m/s*sec15 is so that I can find the final Velocity (cos34*27m/s gives the x component which is constant since there is no acceleration/deceleration)
my answer is .390s which isn't right (ignore sig figs, my teacher turned them off on WA for this assignment)
 
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