Explore how images are used to solve trigonometric equations for sin(x), cos(x), and tan(x) at key points on the unit circle. Understand the solutions visually and gain a deeper understanding of these fundamental trigonometric functions.
Trigonometric Equation
\( \sin(x) = 0 \)
Solutions within \( [0, 2\pi] \):
\( x = 0, \quad x = \pi, \quad x = 2\pi \)
How to interpret the graph:
The graph shows the function \( \sin(x) \) over the interval \( [0, 2\pi] \). The x-axis is marked in terms of \( \pi \). The red dots on the graph represent the points where \( \sin(x) = 0 \). By observing the x-coordinates of these red dots, we can determine the solutions to the equation.
Trigonometric Equation
\( \cos(x) = 0 \)
Solutions within \( [0, 2\pi] \):
\( x = \frac{\pi}{2}, \quad x = \frac{3\pi}{2} \)
How to interpret the graph:
The graph shows the function \( \cos(x) \) over the interval \( [0, 2\pi] \). The x-axis is marked in terms of \( \pi \). When we say \( \cos(x) = 0 \), it’s akin to saying \( y = 0 \) on a standard coordinate system. This means that for the values of \( x \) where \( \cos(x) \) intersects the x-axis, the y-coordinate is zero. The red dots on the graph represent these points of intersection. By observing the x-coordinates of these red dots, we can determine the solutions to the equation. In this case, the cosine function is zero at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), which are the points where the graph intersects the x-axis.
Trigonometric Equation
\( \tan(x) = 0 \)
Solution within \( [-\frac{\pi}{2}, \frac{\pi}{2}] \):
\( x = 0 \)
How to interpret the graph:
The graph shows the function \( \tan(x) \) over the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). The x-axis is marked in terms of \( \pi \). When we say \( \tan(x) = 0 \), it’s akin to saying \( y = 0 \) on a standard coordinate system. This means that for the values of \( x \) where \( \tan(x) \) intersects the x-axis, the y-coordinate is zero. The red dot on the graph represents this point of intersection. By observing the x-coordinate of this red dot, we can determine the solution to the equation. In this case, the tangent function is zero at \( x = 0 \), which is the point where the graph intersects the x-axis.
Trigonometric Equation
\( \tan(x) = 0 \)
Solutions within \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \):
\( x = 0 \)
Explanation:
In the unit circle, the point at \( 0 \) degrees or \( 0 \) radians lies on the x-axis, which means the y-coordinate is \( 0 \). The tangent of \( 0 \) degrees (or \( 0 \) radians) is given by \( \tan(0) = \frac{0}{1} = 0 \). This is why we have \( \tan(x) = 0 \) at \( x = 0 \).
Refer to the unit circle image above for a visual representation.