The next set of identities is the set of
half-angle formulas , which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace
with
the half-angle formula for sine is found by simplifying the equation and solving for
Note that the half-angle formulas are preceded by a
sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which
terminates.
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have
Half-angle formulas
The
half-angle formulas are as follows:
Using a half-angle formula to find the exact value of a sine function
Find
using a half-angle formula.
Since
we use the half-angle formula for sine:
Remember that we can check the answer with a graphing calculator.
Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.
Draw a triangle to represent the given information.
Determine the correct half-angle formula.
Substitute values into the formula based on the triangle.
Simplify.
Finding exact values using half-angle identities
Given that
and
lies in quadrant III, find the exact value of the following:
Using the given information, we can draw the triangle shown in
[link] . Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate
and
Before we start, we must remember that if
is in quadrant III, then
so
This means that the terminal side of
is in quadrant II, since
To find
we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in
[link] and simplify.
We choose the positive value of
because the angle terminates in quadrant II and sine is positive in quadrant II.
To find
we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in
[link] , and simplify.
We choose the negative value of
because the angle is in quadrant II because cosine is negative in quadrant II.
To find
we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in
[link] and simplify.
We choose the negative value of
because
lies in quadrant II, and tangent is negative in quadrant II.
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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