need help with understanding math problem

Question

How is sin(x) = cos(x) the same as tan(x) = 1 and result in x = pi/4 ? Please give the steps to solve for x in sin(x) = cos(x). Thank you.

Answer 1

sinx = cosx

divide both sides by cosx

sinx/cosx = cosx/cosx (cosx/cosx = 1 and sinx/cosx = tanx)

tanx = 1

x = the angle whose tangent = 1, (x = tan^-1(1) )

x = pi/4 = 45 degrees

when you get to tanx = 1, use a calculator with an inverse trig function. plug in 1, then hit inverse tangent, the result is 45 degrees or pi/4

sinx = opposite side over hypotenuse = o/h

cosx - adjacent side over hypotenuse = a/h

tanx = opposite side over adjacent side = o/a = sinx/cosx = (o/h)/(a/h) = (o/h)(h/a) = o/a

when o/a = opposite side over adjacent side = 1, it's a 45 degree angle

tan45 = 1, use a calculator with an inverse trig function

or tan(pi/4) = 1. some calculators give you the answers in degrees some others give answers in radians (pi measures) You can usually switch one to the other, but some come one way or the other, and you may not easily figure out how to switch from one to the other.

tanx =1 involves a fairly common 1-1-sqr2 sided right triangle. It also has angles 45-45-90. After a while you get used to seeing it. It's a right triangle with each of two sides each = 1 and the hypotenuse = sqr2 = about 1.414

It's a triangle with opposite side = height = 1, and adjacent side = base = 1.

tanx = 1 means opposite side/adjacent side = 1/1 = 1. tanx = 1

try graphing it. x=y is a line through the origin. (0,0). run it up to (1,1) That's a line with a 45 degree angle. That's the hypotenuse. construct a right triangle with base = 1. from x=0 to x=1. (0,0) to (1,0) and height = 1 from y=0 to y=1 (1,0) to (1.1)

If you had a protractor, it would show a 45 degree angle.

45 degrees is half way between 0 and 90 degrees. 0 degree is a flat line, horizontal. 90 degrees is a vertical line, perpendicular to the x axis. 45 degrees is half way between flat horizontal and vertical.

but there're also an infinite number of solutions

45 +360n degrees, where n= any integer

and 135 + 360 degrees, where n= integer

or

pi/4 + 2pi(n) and 3pi/4 + 2pi(n) where n= any integer

Answer 2

Hello Dean,

You didn't specify the interval for solving the equation, so I'll assume [0, pi)

You have the equation sin x = cos x

You can divide both sides by cos x and you get:

sin x/cos x = cos x /cos x

on left hand side: What is sin x /cos x ? That's the definition of tan x

on right hand side: What is cos x / cos x? You're dividing something by itself so it is 1.

So the new equation is tan x = 1

1 is a positive number and tan x is positive in Quadrants 1 and 3, so your two solutions will be in Q1 and Q3.

The solution in Q1 in x = pi/4 because on a unit circle, pi/4 is the only Quadrant 1 angle whose sine = its cosine

For the solution in Q3, take pi/4 and ADD pi to it ---> x = pi/4 + pi = 5pi/4

Recap:

sin x = cos x is equivalent to tan x = 1

Solutions in the interval [0, pi) are x = pi/4 and x = 5pi/4