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
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.
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.