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Method and examples
Trigonometry
Method  

2. Find the value of other five trigonometric functions or solve expression
If = in then
 
1. Find the value of other five trigonometric functions
 
2. Solve expression

3. For P(3,4), find the value of all six trigonometric functions
P ( , )
 

4. The Equation for the terminal Side θ is 2x+y=0,x0. Find the value of all six trigonometric functions.
Equation = and
 

Mode =
Decimal Place =

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Find the value of other five trigonometric functions if sin(x)=1/2 calculator
2. Find the value of other five trigonometric functions or solve expression

1. If sin(x)=513 then find other trigonometry

2. If cos(x)=1213 then find other trigonometry

3. If tan(x)=512 then find other trigonometry

4. If csc(x)=135 then find other trigonometry

5. If sec(x)=1312 then find other trigonometry

6. If cot(x)=125 then find other trigonometry


1. If sin(x)=35 then solve cos(x)csc(x)+tan(x)sec(x)

2. If tan(x)=12 then solve 1-tan2(x)1+tan2(x)+2tan(x)1+tan2(x)

3. If sin(x)=45 then solve 1-sin(x)cos(x)+cos(x)1-sin(x)

4. If cos(x)=35 then solve 1-sin(x)cos(x)+cos(x)1-sin(x)



Example
1. If sin(x)=35, find other trigonometry functions sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)

Solution:
sin(x)=35, in Quadrant-1


Opposite side (y), adjacent side (x) and hypotenuse (r)

sin(θ),cos(θ),tan(θ) fromula


sin(θ)=oppositehypotenuse=yr

cos(θ)=adjacenthypotenuse=xr

tan(θ)=oppositeadjacent=yx

csc(θ)=hypotenuseopposite=ry

sec(θ)=hypotenuseadjacent=rx

cot(θ)=adjacentopposite=xy


sin(x)=oppositehypotenuse=yr=35

Here y=3 and r=5

In triangle ABC, by Pythagoras' theorem
r2=x2+y2

:.x^2=r^2-y^2

=5^2-3^2

=25-9

=16

:.x=sqrt(16)=4 (:' x is +ve in Quadrant-1)

So, x=4,y=3 and r=5

(1) sin(x)=y/r=(3)/(5)=3/5

(2) cos(x)=x/r=(4)/(5)=4/5

(3) tan(x)=y/x=(3)/(4)=3/4

(4) csc(x)=r/y=(5)/(3)=5/3

(5) sec(x)=r/x=(5)/(4)=5/4

(6) cot(x)=x/y=(4)/(3)=4/3


Second Method
sin(x)=3/5, in Quadrant-1

(1) cos^2(x)=1-sin^2(x)

=1-(3/5)^2

=1-9/25

=(25-9)/25

=16/25

:.cos(x)=sqrt(16/25)=4/5=4/5


(2) tan(x)=sin(x)/cos(x)=(3/5)/(4/5)=3/5 xx 5/4=3/4=3/4


(3) csc(x)=1/sin(x)=1/(3/5)=5/3=5/3


(4) sec(x)=1/cos(x)=1/(4/5)=5/4=5/4


(5) cot(x)=1/tan(x)=1/(3/4)=4/3=4/3
 




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