-
Home
- Algebra and trigonometry
- The unit circle: sine and cosine
- The other trigonometric functions
Key equations
Tangent function |
|
Secant function |
|
Cosecant function |
|
Cotangent function |
|
Key concepts
- The tangent of an angle is the ratio of the
y -value to the
x -value of the corresponding point on the unit circle.
- The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
- The six trigonometric functions can be found from a point on the unit circle. See
[link]
.
- Trigonometric functions can also be found from an angle. See
[link] .
- Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See
[link] .
- A function is said to be even if
and odd if
for all
x in the domain of
f.
- Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
- Even and odd properties can be used to evaluate trigonometric functions. See
[link] .
- The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
- Identities can be used to evaluate trigonometric functions. See
[link] and
[link]
.
- Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See
[link] .The trigonometric functions repeat at regular intervals.
- The period
of a repeating function
is the smallest interval such that
for any value of
- The values of trigonometric functions can be found by mathematical analysis. See
[link] and
[link]
.
- To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See
[link] .
Section exercises
Verbal
On an interval of
can the sine and cosine values of a radian measure ever be equal? If so, where?
Yes, when the reference angle is
and the terminal side of the angle is in quadrants I and III. Thus, a
the sine and cosine values are equal.
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For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
Substitute the sine of the angle in for
in the Pythagorean Theorem
Solve for
and take the negative solution.
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Tangent and cotangent have a period of
What does this tell us about the output of these functions?
The outputs of tangent and cotangent will repeat every
units.
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Algebraic
For the following exercises, find the exact value of each expression.
For the following exercises, use reference angles to evaluate the expression.
If
and
is in quadrant III, find
and
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Questions & Answers
how does Neisseria cause meningitis
is the branch of biology that deals with the study of microorganisms.
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
How bacteria create energy to survive?
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
which site have a normal flora
Many sites of the body have it
Skin
Nasal cavity
Oral cavity
Gastro intestinal tract
Safaa
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
part of a tissue or an organ being wounded or bruised.
Wilfred
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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