For each of the following equations, use a trigonometric identity to find all solutions.
Using the double-angle formula for
we see that
is a solution of
if and only if
which is true if and only if
To solve this equation, it is important to note that we need to factor the left-hand side and not divide both sides of the equation by
The problem with dividing by
is that it is possible that
is zero. In fact, if we did divide both sides of the equation by
we would miss some of the solutions of the original equation. Factoring the left-hand side of the equation, we see that
is a solution of this equation if and only if
Since
when
and
when
we conclude that the set of solutions to this equation is
Using the double-angle formula for
and the reciprocal identity for
the equation can be written as
To solve this equation, we multiply both sides by
to eliminate the denominator, and say that if
satisfies this equation, then
satisfies the equation
However, we need to be a little careful here. Even if
satisfies this new equation, it may not satisfy the original equation because, to satisfy the original equation, we would need to be able to divide both sides of the equation by
However, if
we cannot divide both sides of the equation by
Therefore, it is possible that we may arrive at extraneous solutions. So, at the end, it is important to check for extraneous solutions. Returning to the equation, it is important that we factor
out of both terms on the left-hand side instead of dividing both sides of the equation by
Factoring the left-hand side of the equation, we can rewrite this equation as
Therefore, the solutions are given by the angles
such that
or
The solutions of the first equation are
The solutions of the second equation are
After checking for extraneous solutions, the set of solutions to the equation is
We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let
be a point on the unit circle and let
be the corresponding angle
Since the angle
and
correspond to the same point
the values of the trigonometric functions at
and at
are the same. Consequently, the trigonometric functions are
periodic functions. The period of a function
is defined to be the smallest positive value
such that
for all values
in the domain of
The sine, cosine, secant, and cosecant functions have a period of
Since the tangent and cotangent functions repeat on an interval of length
their period is
(
[link] ).
Questions & Answers
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?
