f(x)=sin2x and g(x)=1/2tanx for x is the element of [-90;180]
8.1)calculate the x-coordinates of the points of intersection of f and g.
you want
sin(2x) = 1/2 tan(x)
4 sinx cosx = sinx/cosx
sinx (4cos^2(x) - 1) = 0
so, sinx = 0 or cosx = 1/2 or cosx = -1/2
sinx = 0 at x = 0 or 180
cosx = 1/2 at x = -60 or 60
cosx = -1/2 at x = 120
why were you desperate? This is just a little algebra and standard trig angles.
To find the x-coordinates of the points of intersection of f(x) = sin(2x) and g(x) = 1/2tan(x), we can set these two functions equal to each other and solve for x.
f(x) = sin(2x)
g(x) = (1/2)tan(x)
Setting f(x) = g(x):
sin(2x) = (1/2)tan(x)
To solve this equation, we can analyze it step by step:
Step 1: Simplify the equation using trigonometric identities.
sin(2x) = (1/2)sin(x)/cos(x)
Step 2: Multiply both sides of the equation by 2cos(x) to eliminate the fraction.
2cos(x)sin(2x) = sin(x)
Step 3: Use the double-angle identity for sine.
2cos(x)(2sin(x)cos(x)) = sin(x)
Step 4: Distribute and simplify.
4cos^2(x)sin(x) = sin(x)
Step 5: Divide both sides of the equation by sin(x), assuming that sin(x) is not equal to 0.
4cos^2(x) = 1
Step 6: Divide both sides of the equation by 4 and take the square root of both sides.
cos(x) = 1/2 or cos(x) = -1/2
Step 7: Find the values of x that satisfy cos(x) = 1/2 or cos(x) = -1/2 within the given range of [-90, 180].
For cos(x) = 1/2:
x = 60° or x = 300°
For cos(x) = -1/2:
x = 120° or x = 240°
Therefore, the x-coordinates of the points of intersection of f(x) and g(x) within the range [-90, 180] are:
x = 60°, x = 120°, x = 240°, and x = 300°.