On these restricted domains, we can define the
inverse trigonometric functions .
The
inverse sine function
means
The inverse sine function is sometimes called the
arcsine function, and notated
The
inverse cosine function
means
The inverse cosine function is sometimes called the
arccosine function, and notated
The
inverse tangent function
means
The inverse tangent function is sometimes called the
arctangent function, and notated
The graphs of the inverse functions are shown in
[link] ,
[link] , and
[link] . Notice that the output of each of these inverse functions is a
number, an angle in radian measure. We see that
has domain
and range
has domain
and range
and
has domain of all real numbers and range
To find the
domain and
range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line
The sine function and inverse sine (or arcsine) functionThe cosine function and inverse cosine (or arccosine) functionThe tangent function and inverse tangent (or arctangent) function
Relations for inverse sine, cosine, and tangent functions
For angles in the interval
if
then
For angles in the interval
if
then
For angles in the interval
if
then
Writing a relation for an inverse function
Given
write a relation involving the inverse sine.
Finding the exact value of expressions involving the inverse sine, cosine, and tangent functions
Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically
(30°),
(45°), and
(60°), and their reflections into other quadrants.
Given a “special” input value, evaluate an inverse trigonometric function.
Find angle
for which the original trigonometric function has an output equal to the given input for the inverse trigonometric function.
If
is not in the defined range of the inverse, find another angle
that is in the defined range and has the same sine, cosine, or tangent as
depending on which corresponds to the given inverse function.
Evaluating inverse trigonometric functions for special input values
Evaluate each of the following.
Evaluating
is the same as determining the angle that would have a sine value of
In other words, what angle
would satisfy
There are multiple values that would satisfy this relationship, such as
and
but we know we need the angle in the interval
so the answer will be
Remember that the inverse is a function, so for each input, we will get exactly one output.
To evaluate
we know that
and
both have a sine value of
but neither is in the interval
For that, we need the negative angle coterminal with
To evaluate
we are looking for an angle in the interval
with a cosine value of
The angle that satisfies this is
Evaluating
we are looking for an angle in the interval
with a tangent value of 1. The correct angle is
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?
