how to prove sin x=cos(90-x)?

Answer 1

let ABC be a right triangle rt.angled atB
let angle A be x
sinx=opposite side /hypotenuse
=BC/AC
for the complementary angle 90-x the opposite side
becomes the adjacent side
so BC/AC=cos of the angle C
but angle C=90-x
cos(90-x)=sinx

Answer 2

Angles of right triangle are x ,90 degrees and 90-x
and the sides are a,b,c a^2 +b^2 =c^2

sin x = a/c cos x= b/c sin(90-x) =b/c cos(90-x)=a/c

sin x=cos (90-x) = a/c and cos x=sin(90-x)

(sin x)^2 + (cos x)^2 = (a/c)^2 +(b/c)^2 =c^2/c^2 =1

Answer 3

Expand cos(90-x) by the cosine addition formula:
cos(90 -x) = cos 90 cos x + sin 90 sin x.
But
cos 90 = 0
sin 90 = 1
So all you're left with is sin x.

Answer 4

You know that the function sine and cosine are cofunctions, as well as tangent and cotangent, and secant and cosecant. They are pairs.
So how to prove. First, I show you an example of a cofunction: Sin 60deg and Cos 30deg. If you notice, the angles given are complementary, meaning they sum up to 90deg. Using the scientific calculator, if you key in sin 60, you get a value of 0.866...and then key in cos 60, you also get 0.866...meaning if they have the same value, they are cofunctions. Now, the equation you gave is a formula to prove that sine and cosine are cofunctions.
Lets take an example by using the formula. The x there is a variable representing any measure of angle. So lets try x=50. sin x or sin 50=0.766...then cos(90-x)(our x=50, so 90-50=40) so we have cos(90-50) or cos 40. Finally, key in cos 40 and...we also have 0.766...
So to prove that equation, you can give a value of x and then do the equation with the values substituted. If you get the same answer for both sides, then the equation and also the two given trigonometric functions(e.g. sin30 and cos60) are proven.

Answer 5

Visually in the unit circle.

Or using a formula for cos(a-b).
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

cos(90-x) = cos(90)cos(x) + sin(90)sin(x) = sin(x)
Because cos(90) = 0 and sin(90) = 1

Answer 6

you should use compound angle formula:
cos(A - B)=cosAcosB + sinAsinB

cos(90 - x) = cos90 cosx +sin90 sinx
cos90=0 and sinx=1
therefore, cos(90 - x) = sinx (proved)

Answer 7

well u must b knowing that
cos(a-b)= cosa cosb+ sina sinb
using this
cos(90-x)=cos90 cosx+ sin90 sinx
= 0+ sinx
= sinx
simple.