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.)
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
