Sine () and Cosine (): These functions represent the vertical and horizontal coordinates of a point on a unit circle. Both functions have a period of (or radians) and a range of .
Tangent (): Defined as the ratio , the tangent function has a shorter period of (or radians). Its range is infinite, , and it contains vertical asymptotes where the cosine function equals zero.
Key Intercepts: The sine graph passes through the origin , while the cosine graph begins at its maximum value . The tangent graph also passes through the origin but approaches infinity as it nears its asymptotes at .
Periodicity: This is the property where for some constant . For sine and cosine, , meaning the wave repeats every .
Symmetry: The sine function is an odd function, exhibiting rotational symmetry about the origin, such that . The cosine function is an even function, exhibiting reflectional symmetry across the y-axis, such that .
Asymptotic Behavior: In the tangent graph, as approaches from the left, approaches . This occurs because the denominator in approaches zero while the numerator remains positive.
Sketching Step-by-Step: Begin by labeling the x-axis with key intervals (). Mark the maximum and minimum points ( and ) for sine and cosine, or draw vertical dashed lines for tangent asymptotes.
Finding Multiple Solutions: When solving , a calculator provides the principal value. To find other solutions within a range, use the graph's symmetry (e.g., for sine, ) and periodicity (adding/subtracting ).
Applying Transformations: For , the variable affects amplitude, affects the period (), causes a horizontal shift, and causes a vertical shift.
| Feature | Sine () | Cosine () | Tangent () |
|---|---|---|---|
| Period | |||
| Range | |||
| Y-Intercept | |||
| Symmetry | Origin (Odd) | Y-axis (Even) | Origin (Odd) |
| Asymptotes | None | None |
Confusing Period and Frequency: Students often mistake the coefficient in for the period. The period is actually ; a higher value 'squashes' the graph horizontally.
Range Errors: Forgetting that sine and cosine cannot exceed or fall below can lead to attempting to solve impossible equations like .
Tangent Asymptotes: A common error is drawing the tangent curve as a continuous line. It must be drawn as distinct branches separated by asymptotes that the curve never touches.