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

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Solution

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2. Find the value of other five trigonometric functions or solve expression

1. If

sin(x)=513 $sin(x)=5/13$ then find other trigonometry

2. If

cos(x)=1213 $cos(x)=12/13$ then find other trigonometry

3. If

tan(x)=512 $tan(x)=5/12$ then find other trigonometry

4. If

csc(x)=135 $csc(x)=13/5$ then find other trigonometry

5. If

sec(x)=1312 $sec(x)=13/12$ then find other trigonometry

6. If

cot(x)=125 $cot(x)=12/5$ then find other trigonometry

1. If

sin(x)=35 $sin(x)=3/5$ then solve

cos(x)csc(x)+tan(x)sec(x) $cos(x)csc(x)+tan(x)sec(x)$

2. If

tan(x)=12 $tan(x)=1/2$ then solve

1-tan2(x)1+tan2(x)+2tan(x)1+tan2(x) $(1-tan^2(x))/(1+tan^2(x))+(2tan(x))/(1+tan^2(x))$

3. If

sin(x)=45 $sin(x)=4/5$ then solve

1-sin(x)cos(x)+cos(x)1-sin(x) $(1-sin(x))/cos(x)+cos(x)/(1-sin(x))$

4. If

cos(x)=35 $cos(x)=3/5$ then solve

1-sin(x)cos(x)+cos(x)1-sin(x) $(1-sin(x))/cos(x)+cos(x)/(1-sin(x))$

Example

1. If sin(x)=35 $sin(x)=3/5$ , find other trigonometry functions sin(x),cos(x),tan(x),csc(x),sec(x),cot(x) $sin(x),cos(x),tan(x),csc(x),sec(x),cot(x)$

Solution:

sin(x)=35 $sin(x)=3/5$ , in Quadrant-1

Opposite side

(y) $(y)$ , adjacent side

(x) $(x)$ and hypotenuse

(r) $(r)$

sin(θ),cos(θ),tan(θ) $sin(theta), cos(theta), tan(theta)$ fromula

sin(θ)=oppositehypotenuse=yr $sin(theta) = "opposite"/"hypotenuse" = y/r$

cos(θ)=adjacenthypotenuse=xr $cos(theta) = "adjacent"/"hypotenuse" = x/r$

tan(θ)=oppositeadjacent=yx $tan(theta) = "opposite"/"adjacent" = y/x$

csc(θ)=hypotenuseopposite=ry $csc(theta) = "hypotenuse"/"opposite" = r/y$

sec(θ)=hypotenuseadjacent=rx $sec(theta) = "hypotenuse"/"adjacent" = r/x$

cot(θ)=adjacentopposite=xy $cot(theta) = "adjacent"/"opposite" = x/y$

sin(x)=oppositehypotenuse=yr=35 $sin(x) = "opposite"/"hypotenuse" = y/r = 3/5$

Here

y=3 $y=3$ and

r=5 $r=5$

In triangle ABC, by Pythagoras' theorem

r2=x2+y2 $r^2=x^2+y^2$

∴x2=r2-y2 $:.x^2=r^2-y^2$

=52-32 $=5^2-3^2$

=25-9 $=25-9$

=16 $=16$

∴x=√16=4 $:.x=sqrt(16)=4$ (

∵ $:'$ x is +ve in Quadrant-1)

So,

x=4,y=3andr=5 $x=4,y=3 and r=5$

(1) $(1)$

sin(x)=yr=35=35 $sin(x)=y/r=(3)/(5)=3/5$

(2) $(2)$

cos(x)=xr=45=45 $cos(x)=x/r=(4)/(5)=4/5$

(3) $(3)$

tan(x)=yx=34=34 $tan(x)=y/x=(3)/(4)=3/4$

(4) $(4)$

csc(x)=ry=53=53 $csc(x)=r/y=(5)/(3)=5/3$

(5) $(5)$

sec(x)=rx=54=54 $sec(x)=r/x=(5)/(4)=5/4$

(6) $(6)$

cot(x)=xy=43=43 $cot(x)=x/y=(4)/(3)=4/3$

Second Method

sin(x)=35 $sin(x)=3/5$ , in Quadrant-1

(1) $(1)$

cos2(x)=1-sin2(x) $cos^2(x)=1-sin^2(x)$

=1-(35)2 $=1-(3/5)^2$

=1-925 $=1-9/25$

=25-925 $=(25-9)/25$

=1625 $=16/25$

∴cos(x)=√1625=45=45 $:.cos(x)=sqrt(16/25)=4/5=4/5$

(2) $(2)$

tan(x)=sin(x)cos(x)=3545=35×54=34=34 $tan(x)=sin(x)/cos(x)=(3/5)/(4/5)=3/5 xx 5/4=3/4=3/4$

(3) $(3)$

csc(x)=1sin(x)=135=53=53 $csc(x)=1/sin(x)=1/(3/5)=5/3=5/3$

(4) $(4)$

sec(x)=1cos(x)=145=54=54 $sec(x)=1/cos(x)=1/(4/5)=5/4=5/4$

(5) $(5)$

cot(x)=1tan(x)=134=43=43 $cot(x)=1/tan(x)=1/(3/4)=4/3=4/3$

Input functions

Sr No.	Function	Input value
1.	x3 $x^3$	x^3
2.	√x $sqrt(x)$	sqrt(x)
3.	3√x $root(3)(x)$	root(3,x)
4.	sin(x)	sin(x)
5.	cos(x)	cos(x)
6.	tan(x)	tan(x)
7.	sec(x)	sec(x)
8.	cosec(x)	csc(x)
9.	cot(x)	cot(x)
10.	sin-1(x) $sin^(-1)(x)$	asin(x)
11.	cos-1(x) $cos^(-1)(x)$	acos(x)
12.	tan-1(x) $tan^(-1)(x)$	atan(x)
13.	sin2(x) $sin^2(x)$	sin^2(x)
14.	logy(x) $log_y(x)$	log(y,x)
15.	log10(x) $log_10(x)$	log(x)
16.	loge(x) $log_e(x)$	ln(x)
17.	ex $e^x$	exp(x) or e^x
18.	e2x $e^(2x)$	exp(2x) or e^(2x)
19.	∞ $oo$	inf