The graphs of sin(x) and cos(x) intersect at points where sin(x) = cos(x). This occurs at x = π/4 + nπ, where n is an integer.
Finding the Intersection Points
To find where the sine and cosine graphs intersect, we need to solve the equation sin(x) = cos(x).
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Divide by cos(x): Assuming cos(x) ≠ 0, we can divide both sides of the equation by cos(x) to get tan(x) = 1.
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Solve for x: The tangent function equals 1 at π/4 radians (45 degrees) and repeats every π radians (180 degrees). Therefore, the general solution is x = π/4 + nπ, where n is an integer.
Examples within the interval [0, 2π]
Within the interval of 0 to 2π (a full circle), the sine and cosine graphs intersect at two points:
- x = π/4: This is the first intersection point in the first quadrant. sin(π/4) = cos(π/4) = √2/2.
- x = 5π/4: This is the second intersection point in the third quadrant. sin(5π/4) = cos(5π/4) = -√2/2.
Graphical Representation
Visually, you can see these intersections by graphing y = sin(x) and y = cos(x). The points where the two curves cross each other represent the solutions to the equation sin(x) = cos(x). The provided reference video confirms that, within a given range (0 to 2π), there are typically two points of intersection.
