Please help me understand this trig. problem that would require a drawing on a grid paper.
An alternative way to define the sine,cosine,tangent of an angle is by using a circle of radius one centered at the origin of a coordinate system.
Let (x,y) be the point on the unit circle that corresponds to an angle A.Then the sine,cosine and tangent of angle A are defined as follows:
Sin A=y
Cos A=x
tan A=y/x,=/=0
An alternative way to define the sine,cosine,tangent of an angle is by using a circle of radius one centered at the origin of a coordinate system.
Let (x,y) be the point on the unit circle that corresponds to an angle A.Then the sine,cosine and tangent of angle A are defined as follows:
Sin A=y
Cos A=x
tan A=y/x,=/=0
-
If you draw a circle of radius = 1, centered at the origin (0, 0) that circle is the unit circle.
Its is called the UNIT circle because the radius is 1 unit and its center is the origin.
for any angle the end of the radius is the (x, y) point being talked about
the angle of 0º is defined to be the positive x axis, the end of radius is the (x, y).
That point is (1,0)
What is cos (0º)? what is sin (0º)?
cos (0º) = 1 and sin (0º) = 0
we rotate around the circle counter clockwise
The angle of 90º is the positive y axis, and its endpoint is (0, 1)
What is cos (90º)? what is sin (90º)?
cos (90º) = 0 sin(90º) = 1
Now consider the 45º angle. The end of the radius at the angle is (√2/2, √2/2)
What is cos (45º)? what is sin (45º)?
cos (45º) = sin (45º) = √2/2
as I rotate past 90º, I change quadrants.
My x value will now be negative, it will go from 0 to - 1
my y values will go from 1 to 0
an angle of 180º will be half way around the circle.
The radius will have an endpoint of (- 1, 0)
Now let me make a large rotation, say 210º
That is 30º more than 180º
That rotation will take me into quad III, with a reference angle of 30º, below or beyond the 180º rotation
the end of that radius wil be (-1/2, -√3/2)
both x and y values are going to be negative in quad III
In general, you draw a unit circle, and rotate the specified angle, call it T.
At the end of that radius, is a point (x, y) where x = cos (T) and y = sin (T)
To envision the rt triangle, most people drop a perpendicular from the endpoint of that radius to the x axis. The perpendicular is vertical. Now you have a right triangle with the radius being the hypotenuse = 1, the sine of that angle, is the length of the vertical leg, or y and the cos of that angle is the length of the horizontal leg, or x
Its is called the UNIT circle because the radius is 1 unit and its center is the origin.
for any angle the end of the radius is the (x, y) point being talked about
the angle of 0º is defined to be the positive x axis, the end of radius is the (x, y).
That point is (1,0)
What is cos (0º)? what is sin (0º)?
cos (0º) = 1 and sin (0º) = 0
we rotate around the circle counter clockwise
The angle of 90º is the positive y axis, and its endpoint is (0, 1)
What is cos (90º)? what is sin (90º)?
cos (90º) = 0 sin(90º) = 1
Now consider the 45º angle. The end of the radius at the angle is (√2/2, √2/2)
What is cos (45º)? what is sin (45º)?
cos (45º) = sin (45º) = √2/2
as I rotate past 90º, I change quadrants.
My x value will now be negative, it will go from 0 to - 1
my y values will go from 1 to 0
an angle of 180º will be half way around the circle.
The radius will have an endpoint of (- 1, 0)
Now let me make a large rotation, say 210º
That is 30º more than 180º
That rotation will take me into quad III, with a reference angle of 30º, below or beyond the 180º rotation
the end of that radius wil be (-1/2, -√3/2)
both x and y values are going to be negative in quad III
In general, you draw a unit circle, and rotate the specified angle, call it T.
At the end of that radius, is a point (x, y) where x = cos (T) and y = sin (T)
To envision the rt triangle, most people drop a perpendicular from the endpoint of that radius to the x axis. The perpendicular is vertical. Now you have a right triangle with the radius being the hypotenuse = 1, the sine of that angle, is the length of the vertical leg, or y and the cos of that angle is the length of the horizontal leg, or x
-
What needs explanation?
The sine is defined as (opposite)/(hypotenuse).
When the hypotenuse is a radius to a unit circle, it is constant and set to 1. Thus, (hypotenuse) falls out of the equation, and sine = (opposite). In the case of a unit circle centered on the origin, (opposite) = y.
The others follow by the same kind of argument.
The sine is defined as (opposite)/(hypotenuse).
When the hypotenuse is a radius to a unit circle, it is constant and set to 1. Thus, (hypotenuse) falls out of the equation, and sine = (opposite). In the case of a unit circle centered on the origin, (opposite) = y.
The others follow by the same kind of argument.