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.
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.