-
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.
Got questions? Get instant answers now!
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.
Got questions? Get instant answers now!
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.
Got questions? Get instant answers now!
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
Got questions? Get instant answers now!
Questions & Answers
if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
hey , can you please explain oxidation reaction & redox ?
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
advantages of electrons in a circuit
we're do you find electromagnetism past papers
Ntombifuthi
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is the half reaction of Potassium and chlorine
how to calculate coefficient of static friction
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
How is energy being used in bonding?
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
how to calculate tension by substitution
use fnet method. how many obects are being calculated ?
Khuthadzo
how to calculate acceleration and tension force
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
how to calculate normal force
Mulaudzi
Got questions? Join the online conversation and get instant answers!
Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.