Name:
Date:
Period:
Trig Identities Performance Task
The Physics of Soccer: How would you have to kick a soccer ball so that it would go as far as you could
possibly kick it? Hypothesize what elements are in your control (and what are not). Describe how you think
you could attain maximum distance.
The formula for the maximum height h in feet of a projectile is ℎ =
(𝑣0 )2 (
tan 𝜃 − sin 𝜃 cos 𝜃
)
tan 𝜃
, where v0 is the
2𝑔
initial velocity in feet per second, θ is the measure of the angle of elevation in degrees, and g is the
acceleration due to gravity in feet per second squared.

Write a new formula for h by simplifying
cosine.
tan 𝜃 − sin 𝜃 cos 𝜃
tan 𝜃
. Rewrite the fraction ONLY using sine and
Equation: h=
The distance d that a projected object travels, in feet, is given by the formula 𝑑 =
1
2(𝑣0 )2
1
tan 𝜃 + cot 𝜃
𝑔
.
Write a new formula for d by simplifying tan 𝜃 + cot 𝜃. Rewrite the fraction ONLY using sine and cosine.
Equation: d=
QUESTIONS:
1. A soccer player kicks a ball at an initial velocity of 60 feet per second at an angle of 40°. The
acceleration due to gravity is 32 ft/s 2 . Find the maximum height reached by the ball. Round to the
nearest tenth. (show your work)
2. With what initial velocity must a soccer player kick a ball at an angle of 35° in order for it to reach a
maximum height of 20 feet?
3. Find the distance traveled by the ball in question 1.
In order to kick a ball the greatest possible distance at a given initial velocity, a soccer player must
maximize the distance equation. Since 2, 𝑣0 , and g are constants, this means the player must
maximize the sine and cosine values. Use the patterns in the table to hypothesize a value of θ for
which sin θ cos θ will be maximal. Use a calculator to check your hypothesis. At what angle should
the player kick the ball to achieve the greatest distance?
4.
Angle (𝜃)
0
10
Calculation
sin 0° cos 0° = sin 90° cos 90°
sin 10° cos 10° = sin 80° cos 80°
Value
0
0.1710
5. Determine the maximum distance the soccer player from question 1 could kick the ball.