Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of
formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If
for higher-level competition, what is the measurement of the angle for novice competition?
Since the angle for novice competition measures half the steepness of the angle for the high level competition, and
for high competition, we can find
from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See
[link] .
We see that
We can use the half-angle formula for tangent:
Since
is in the first quadrant, so is
We can take the inverse tangent to find the angle:
So the angle of the ramp for novice competition is
Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See
[link] ,
[link] ,
[link] , and
[link] .
Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See
[link] and
[link] .
Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Explain how to determine the reduction identities from the double-angle identity
Use the Pythagorean identities and isolate the squared term.
We can determine the half-angle formula for
by dividing the formula for
by
Explain how to determine two formulas for
that do not involve any square roots.
multiplying the top and bottom by
and
respectively.
For the half-angle formula given in the previous exercise for
explain why dividing by 0 is not a concern. (Hint: examine the values of
necessary for the denominator to be 0.)
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
