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
$\text{\hspace{0.17em}}f(-x)=f(x)\text{\hspace{0.17em}}$ and odd if
$\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).$
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
$\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of a repeating function
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the smallest interval such that
$\text{\hspace{0.17em}}f(x+P)=f(x)\text{\hspace{0.17em}}$ for any value of
$\text{\hspace{0.17em}}x.$
The values of trigonometric functions of special angles can be found by mathematical analysis.
To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See
[link] .
Section exercises
Verbal
On an interval of
$\text{\hspace{0.17em}}\left[0,2\pi \right),$ can the sine and cosine values of a radian measure ever be equal? If so, where?
Yes, when the reference angle is
$\text{\hspace{0.17em}}\frac{\pi}{4}\text{\hspace{0.17em}}$ and the terminal side of the angle is in quadrants I and III. Thus, at
$\text{\hspace{0.17em}}x=\frac{\pi}{4},\frac{5\pi}{4},$ the sine and cosine values are equal.
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
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in the Pythagorean Theorem
$\text{\hspace{0.17em}}{x}^{2}+{y}^{2}=1.\text{\hspace{0.17em}}$ Solve for
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and take the negative solution.
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.