Transcript tan q - ClassZone

EXAMPLE 4
Derive a trigonometric model
Soccer
Write an equation for the horizontal distance traveled
by a soccer ball kicked from ground level (h0 = 0) at
speed v and angle q.
EXAMPLE 4
Derive a trigonometric model
SOLUTION
16
2 + (tan q )x + 0 = 0
– 2
x
v cos2 q
– x(
Let h0 = 0.
16
x – tan q ) = 0
2
2
v cos q
Factor.
16
x – tan q = 0
2
2
v cos q
Zero product property
16
x = tan q
2
2
v cos q
Add tan q to each side.
1 2
x = 16 v cos2 q tan q
Multiply each side by
1 2
2q.
v
cos
16
EXAMPLE 4
Derive a trigonometric model
x=
1 2
16 v cos q sin q
Use cos q tan q = sin q.
1
1
1 2
Rewrite
as
2.
=
x 32 v (2cos q sin q )
16
32
1
x = 32 v2 sin 2q
Use a double-angle formula.
GUIDED PRACTICE
for Examples 4
9. WHAT IF? Suppose you kick a soccer ball from
ground level with an initial speed of 70 feet per
second. Can you make the ball travel 200 feet?
ANSWER
No
1 2
10. REASONING: Use the equation x = 32 v sin 2 q to
explain why the projection angle that maximizes
the distance a soccer ball travels is q = 45°.
ANSWER
sin 2(45°) = sin 90° = 1, this is the only value that will
not decrease the velocity of the ball since it is the
only non-fractional value.