Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
Using the unit circle, the sine of an angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the
y -value of the endpoint on the unit circle of an arc of length
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ whereas the cosine of an angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the
x -value of the endpoint. See
[link] .
The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See
[link] .
When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See
[link] .
Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See
[link] .
The domain of the sine and cosine functions is all real numbers.
The range of both the sine and cosine functions is
$\text{\hspace{0.17em}}[-1,1].\text{\hspace{0.17em}}$
The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
The signs of the sine and cosine are determined from the
x - and
y -values in the quadrant of the original angle.
An angle’s reference angle is the size angle,
$\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis. See
[link] .
Reference angles can be used to find the sine and cosine of the original angle. See
[link] .
Reference angles can also be used to find the coordinates of a point on a circle. See
[link] .
Section exercises
Verbal
Describe the unit circle.
The unit circle is a circle of radius 1 centered at the origin.
Discuss the difference between a coterminal angle and a reference angle.
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle,
$\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle
$\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis.
For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by
$t$ lies.
$\text{sin}(t)<0\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\text{cos}(t)<0\text{\hspace{0.17em}}$
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.