Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. The trigonometric functions are functions of an angle. It is also called as circular function. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The familiar trigonometric functions are the sine, cosine, and tangent. The other trigonometric function like cosecant, secant and cot are related to the familiar function.
Source Wikipedia.
Some Trigonometric related functions:
1. `sin^2 theta + cos^2 theta = 1`
2. `tan theta = sin theta / cos theta`
3.` cot theta = 1/tan theta = cos theta/ sin theta `
4. `1+ tan^2 theta = sec^2 theta`
5.`sec theta ` = `1/cos theta`
6.` cosec theta = 1/ sin theta`
7. `sin 2theta = 2 sin theta cos theta`
8.`sin^2 theta ` = `(1 - cos 2theta)/2`
More details about trigonometric functions:
In Right-angle triangle
Sine (Sin):
In right-angle triangle, Ratio of the opposite side length and the hypotenuse of an angle is called as sine. It is reciprocal of cosecant.
` sin theta` = (opposite side) / (hypotenuse side)
Tangent (Tan):
In right-angle triangle, Ratio of the opposite side length and the adjacent side length of an angle is called as tangent. In trigonometry relation is the ratio of sin and cosine. The tangent is reciprocal of cot.
` tan theta` = (opposite side) / (adjacent side)
Secant (Sec) = `1/cos` :
In right-angle triangle, Ratio of the hypotenuse and the adjacent side length of an angle is called as secant. The secant is reciprocal of cosine.
`Sec theta` = (hypotenuse)/(adjacent) <br>
Cosine (Cos):
In right-angle triangle, Ratio of the adjacent side length and the hypotenuse of an angle is called as cosine. It is a reciprocal of secant.
.`cos theta` = (adjacent side)/(hypotenuse)
Cosecant (cosec) = `1/sin` :
In right-angle triangle, Ratio of the hypotenuse and the opposite side length of an angle is called as cosecant. The cosecant is reciprocal of sine.
` cosec theta` = (hypotenuse)/(opposite side)
Cot` (1/ tan)` :
In right-angle triangle, Ratio of the adjacent side length and the opposite side of an angle is called as cot. In trigonometry relation is the ratio of cosine and sin. The cot is reciprocal of tangent.
`cot theta ` = (adjacent side)/(opposite side)
Trigonometric problems:
Trigonometric problem 1:
If x = a cos t + b sin t and y = a sin t − b cos t, show that x2 + y2 = a2 + b2
Solution:
x2 + y2 = (a cos t+ b sin t)2 + (a sin t − b cos t)2
= a2 cos2t + b2 sin2t + 2ab cos t sin t+ a2 sin2t + b2 cos2t − 2ab sint cost
= a2 (cos2t + sin2t) + b2(sin2t + cos2t)+ 2ab cos t sin t − 2ab sint cost
= a2 + b2
Hence it has been proved.
Trigonometric problem 2:
Prove that trigonometric relation: `(sec theta + 1/ (cot theta))^2 ` = `(1 + sin theta)/(1 - sin theta)`
Solution:
Take left side, `sec theta + 1/cot theta `
Squared the term, `sec theta + 1/cot theta` = `(sec theta + 1/cot theta)^2`
Now,
`(sec theta + 1/cot theta)^2 = ( (1/cos theta) + (sin theta / cos theta) )^2` ; we know, `sec theta = (1/cos theta) ; 1/cot theta = (sin theta /cos theta)`
= `(1 + sin theta )^2/ (cos^2theta)`
= `(1 + sin theta)^2 / (1 - sin^2 theta)` ; sin2θ + cos2θ = 1; so,cos2θ = 1- sin2θ
= `((1 + sin theta)(1 + sin theta)) / ((1 + sin theta) (1 - sin theta))`
`(sec theta + tan theta)^2` = `(1 + sin theta)/(1 - sin theta)`
= Right side.
Hence, the given trigonometric relation is proved.
No comments:
Post a Comment