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
Thus,
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.)
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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