Given f(x)=1+sinX and get g(x)=cos 2X
Calculate the points of intersection of the graph f and g for x€{180°;360°}
f(x)=g(x)
1+sinx=cos2x=1-sin^2 x
change variable sinx = u
1+u=1-u^2
u^2+u=0
u=0 or u=-1
or x=0deg, 180 deg or x=270 (or -90deg).
cos(2X) = 1 - 2 sin^2(X)
1 + sin(X) = 1 - 2 sin^2(X)
solve for sin(X) , match solutions for x€{180°;360°}
To find the points of intersection between the graphs of f(x) = 1 + sin(x) and g(x) = cos(2x) for x ∈ {180°, 360°}, we will set the two equations equal to each other and solve for x.
First, let's write the equation:
1 + sin(x) = cos(2x)
Now, we need to simplify the equation. We know that cos(2x) can be written in terms of sin(x) using the double-angle identity for cosine:
cos(2x) = cos^2(x) - sin^2(x)
Now, we can substitute this into our equation:
1 + sin(x) = cos^2(x) - sin^2(x)
To simplify further, let's use the Pythagorean identity for sine and cosine:
sin^2(x) + cos^2(x) = 1
Substituting this into our equation:
1 + sin(x) = 1 - sin^2(x) - sin^2(x)
Now, we can combine like terms:
1 + sin(x) = 1 - 2sin^2(x)
Rearranging the equation, we get:
2sin^2(x) + sin(x) - 1 = 0
Now, this is a quadratic equation in terms of sin(x). Let's solve it by factoring or using the quadratic formula.
2sin^2(x) + sin(x) - 1 = 0
Factoring the quadratic equation, we get:
(2sin(x) - 1)(sin(x) + 1) = 0
Setting each factor equal to zero and solving for sin(x), we have two equations:
2sin(x) - 1 = 0
sin(x) + 1 = 0
Solving the first equation:
2sin(x) - 1 = 0
2sin(x) = 1
sin(x) = 1/2
Using the unit circle or a calculator, we find that sin(x) = 1/2 when x = 30° or x = 150°.
Now, solving the second equation:
sin(x) + 1 = 0
sin(x) = -1
x = 180°
Therefore, the three points of intersection for x ∈ {180°, 360°} are:
1) x = 30°, sin(x) = 1/2
2) x = 150°, sin(x) = 1/2
3) x = 180°, sin(x) = -1
Note that we obtained the sine values and found the corresponding x values from the unit circle or by using a calculator.
To find the points of intersection between the graphs of f(x) = 1 + sin(x) and g(x) = cos(2x) for x ∈ [180°, 360°], we need to set both functions equal to each other and solve for x.
First, let's set the two functions equal to each other:
1 + sin(x) = cos(2x)
Next, let's simplify the equation:
1 + sin(x) = cos^2(x) - sin^2(x)
Using the trigonometric identity cos^2(x) - sin^2(x) = cos(2x), we can rewrite the equation as:
1 + sin(x) = cos(2x)
Now, let's solve for x. However, solving this equation algebraically is quite complex and may involve higher order trigonometric identities. Instead, let's use a graphical approach or numerical methods to find the points of intersection.
Graphical Approach:
1. Plot the graphs of f(x) = 1 + sin(x) and g(x) = cos(2x) on the same coordinate system.
2. Identify the x-values where the graphs intersect within the range x ∈ [180°, 360°]. These are the points of intersection.
Numerical Approach:
1. Choose a small step size (e.g., 0.01) within the range x ∈ [180°, 360°].
2. Calculate the corresponding values of f(x) = 1 + sin(x) and g(x) = cos(2x) for each x-value using a calculator or software.
3. Look for x-values where the calculated values of f(x) and g(x) are approximately equal. These are the points of intersection.
By either of these approaches, you can determine the points of intersection between the graphs of f(x) = 1 + sin(x) and g(x) = cos(2x) for x ∈ [180°, 360°].