Two trigometric functions intersect point

In summary, the conversation revolved around finding the intersection of the graphs of y = tan x and y = √2 * cos x without using a calculator. The discussion also explored the possibility of finding a solution using trigonometric properties and suggested constructing isosceles and 30-60-90 triangles to aid in solving the problem. Ultimately, the conversation concluded with the suggestion of converting all terms to sin(x) to create a quadratic equation and solving for a value of x within the given range.
  • #1
likeachild
7
0

Homework Statement



I am trying to find out how to solve for x without a calculator.
Basically where [tex] tan({x}) [/tex] and [tex]sqrt{2}*cos{x}[/tex] intersect.

Homework Equations



Find [tex]x[/tex] in the range of [tex]0 \le {x} \le \frac {pi}{2}[/tex]

The Attempt at a Solution



I couldn't find out how to solve this without a calculator.
I tried fooling around with the trigometric properties like the double argument and pythagorean, but I still couldn't find out.
My teacher doesn't know either. lol.

The answer by looking at it graphically is [tex]\frac {pi}{4}[/tex]
 
Last edited:
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  • #2
sin(x)=sqrt(2)cos^2(x)

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

sin(x)=sqrt(2)[1-sin^2(x)]

Quadratic eqn- solve for sin(x)
 
  • #3
Doing without a calculator hints that the solution might be something like 30 deg, or 45 deg, or 90 deg, etc., an angle whose sin, cos, tan you should have memorized. Let's try that...

There are 2 right-angled triangles you need to be able to sketch without even thinking:-

1) an isosceles triangle with base angles of 45 deg. (label the sides 1,1, and root something)
2) a triangle with angles of 30, 60 and 90 degrees, and sides of 1,2, and root something

Construct them. These allow you to, by inspection, write down equations for sin 45, sin 60, tan 30, tan 45, and so on.

Good luck!
 
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  • #4
likeachild said:

Homework Statement



I am trying to find out how to solve for x without a calculator.
Basically where [tex] tan({x}) [/tex] and [tex]sqrt{2}*cos{x}[/tex] intersect.

Since the two items noted are not formulae, as I understand it they can't intersect. What did you really mean? Are the values equal?
 
  • #5
AC130Nav said:
Since the two items noted are not formulae, as I understand it they can't intersect.
Question concerns two graphs,
viz., y = tan x
and y = root2 * cos x

for x between 0 and Pi / 2 the curves intersect at one point.
 
  • #6
Your equation is
[tex]\frac{sin(x)}{cos(x)}= \sqrt{2}cos(x)[/tex]

Convert ever thing to sin(x) and you will have quadratic equation in sin(x).
 
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What are the two trigonometric functions that intersect at a point?

The two trigonometric functions that intersect at a point are sine (sin) and cosine (cos).

How can you find the point of intersection between two trigonometric functions?

To find the point of intersection between two trigonometric functions, set the two functions equal to each other and solve for the variable. The resulting value will be the x-coordinate of the point of intersection. Then, plug this value into either of the original equations to find the y-coordinate.

What is the significance of the point of intersection between two trigonometric functions?

The point of intersection between two trigonometric functions represents the solution to a system of equations involving those functions. It is also the point where the graphs of the two functions intersect on the coordinate plane.

Can two trigonometric functions intersect at more than one point?

Yes, two trigonometric functions can intersect at more than one point. This will occur when the two functions have multiple solutions for the same value of x when set equal to each other.

How can you graph the point of intersection between two trigonometric functions?

To graph the point of intersection between two trigonometric functions, plot the x and y coordinates of the point on the coordinate plane. Then, draw a small circle or dot at that point to represent the intersection. You can also use the graphing calculator to plot the functions and find the coordinates of the point of intersection.

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