Understand Trigonometric Functions. Click on the link to Watch the VIDEO explanation:
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Trigonometric Functions
We shall understand the meaning of Trigonometric Functions and the important Trigonometric Functions
Trigonometrical Functions are all functions of theta because for each angle theta they exist exactly one value of the ratio. Trigonometric Functions are also called the cellular functions. The figure shows a graph in the first quadrant. Let O be the origin, let OX be the x axis and OY be the y axis,
let theta be the any real number. Construct the angle whose measures is theta radians, with vertex at the origin of a rectangular coordinate system and initial side along the positive x - axis. Let P (x, y) be any point on the terminal side OA of the angle. Let OP is equal to r.
OM is equal to x and PM is equal to y. Triangle OPM is the right angled triangle.
The ratio y by r is called the sine of theta and is written as sin theta. Therefore sin theta is equal to y by r
The ratio x by r is called the cosine of theta and is written as cos theta. Therefore cos theta is equal to x by r.
Tan theta is equal to y by x provided theta is not an odd multiple of pie by 2 i.e., theta should not be equal to 3 pie by 2 , 5 pie by 2 etc.
The ratio x by y is called the cotangent of theta and is written as cot theta, provided theta is not an even multiple of pie by 2 that is the values of the theta should not be equal 2 pie, 6 pie etc.
Secant theta is equal to r by x provided theta is not an odd multiple of pie by 2.
Cosecant theta is equal to r by y provided theta is not an even multiple of pie by 2.
These functions sin theta, cos theta, tan theta, cot theta, sec theta, cosec theta are called trigonometric Functions. Click on the next button.
Now, we show that these trigonometric functions are well - defined and their values depends only on the value of theta and not on the position of point P on the terminal side.
Let P dash (x dash, y dash) be any other point on the terminal side OA. Let OP dash is equal to r dash.
Draw a perpendicular from P dash to OX at M dash.
The triangles OMP and OM dash P dash are similar.
Therefore, M dash P dash by OP dash is equal to MP by OP
This implies mode of y dash by r dash is equal to mode of y by r.
y dash by r dash is equal to y by r Since P and P dash are in the same quadrant mode of y dash is equal to y dash and mode of y is equal to y.
If OA is along x axis, then we have y by r is equal to 0 by r therefore is equal to 0
If OA is along y axis, then we have y by r is equal to r by r.
Therefore, the ratio y by r depends only upon the values of theta and not on the position of P.
Click on the next button.
Some of the possible positions of the terminal side of the angle theta radians are as given in the following figures.
Watch Video
Trigonometric Functions
We shall understand the meaning of Trigonometric Functions and the important Trigonometric Functions
Trigonometrical Functions are all functions of theta because for each angle theta they exist exactly one value of the ratio. Trigonometric Functions are also called the cellular functions. The figure shows a graph in the first quadrant. Let O be the origin, let OX be the x axis and OY be the y axis,
let theta be the any real number. Construct the angle whose measures is theta radians, with vertex at the origin of a rectangular coordinate system and initial side along the positive x - axis. Let P (x, y) be any point on the terminal side OA of the angle. Let OP is equal to r.
OM is equal to x and PM is equal to y. Triangle OPM is the right angled triangle.
The ratio y by r is called the sine of theta and is written as sin theta. Therefore sin theta is equal to y by r
The ratio x by r is called the cosine of theta and is written as cos theta. Therefore cos theta is equal to x by r.
Tan theta is equal to y by x provided theta is not an odd multiple of pie by 2 i.e., theta should not be equal to 3 pie by 2 , 5 pie by 2 etc.
The ratio x by y is called the cotangent of theta and is written as cot theta, provided theta is not an even multiple of pie by 2 that is the values of the theta should not be equal 2 pie, 6 pie etc.
Secant theta is equal to r by x provided theta is not an odd multiple of pie by 2.
Cosecant theta is equal to r by y provided theta is not an even multiple of pie by 2.
These functions sin theta, cos theta, tan theta, cot theta, sec theta, cosec theta are called trigonometric Functions. Click on the next button.
Now, we show that these trigonometric functions are well - defined and their values depends only on the value of theta and not on the position of point P on the terminal side.
Let P dash (x dash, y dash) be any other point on the terminal side OA. Let OP dash is equal to r dash.
Draw a perpendicular from P dash to OX at M dash.
The triangles OMP and OM dash P dash are similar.
Therefore, M dash P dash by OP dash is equal to MP by OP
This implies mode of y dash by r dash is equal to mode of y by r.
y dash by r dash is equal to y by r Since P and P dash are in the same quadrant mode of y dash is equal to y dash and mode of y is equal to y.
If OA is along x axis, then we have y by r is equal to 0 by r therefore is equal to 0
If OA is along y axis, then we have y by r is equal to r by r.
Therefore, the ratio y by r depends only upon the values of theta and not on the position of P.
Click on the next button.
Some of the possible positions of the terminal side of the angle theta radians are as given in the following figures.
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