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Cosine is one of the three main trigonometric functions, along with sine and tangent. These functions are used to explore relationships within a right-angled triangle and extend to modeling periodic phenomena. The cosine function, denoted as \( \cos \theta \), outputs values based on the angle \( \theta \), typically measured in radians. When graphed, the function forms a wave-like pattern that oscillates smoothly between -1 and 1.

Key characteristics of the cosine graph include:

• It starts at its maximum value at \( \theta = 0 \), unlike the sine function, which starts at zero.
• The graph completes one full wave, or cycle, over an interval of \( 2\pi \) radians.
• Like all trigonometric graphs, it repeats indefinitely, showing periodic behavior.

Understanding these characteristics is crucial to sketch any trigonometric graph, as well as transform and adjust them as the equations become more complex.

Amplitude refers to the height of the wave created by a trigonometric function like cosine. In its simplest form, the graph of \( y = \cos \theta \) has an amplitude of 1, indicating the wave peaks at 1 and dips at -1.

When a coefficient multiplies the function, such as the \( -3 \) in \( y = -3 \cos \theta \), it changes the amplitude. In this case, the amplitude becomes 3, meaning the graph stretches or compresses vertically to reach 3 and -3 instead of 1 and -1.

Important points about amplitude:

• The amplitude determines how "tall" or "short" the trigonometric wave will appear.
• A negative sign before the amplitude inverts the graph, flipping it upside down. Thus, the graph begins at \( -3 \) when \( \theta = 0 \).
• While it affects height, the amplitude does not alter the period or frequency of the wave's cycle.

Recognizing how the amplitude modifies the graph is essential for accurately sketching and interpreting these equations.

Graph transformations involve shifting, stretching, compressing, and reflecting the base graphs of functions. In the case of the cosine function, transformations help to adapt the basic wave to fit new requirements as dictated by function manipulation.

For \( y = -3 \cos \theta \), two transformations are applied:

• The amplitude change to 3, which stretches the graph vertically compared to the standard cosine function.
• The negative sign which reflects the graph across the horizontal axis, resulting in an 'inverted' wave.

These actions ensure the graph starts at \( -3 \) at \( \theta = 0 \) and peaks at this value, switches to a minimum point of 3 at \( \theta = \pi \), and returns to \( -3 \) at \( \theta = 2\pi \).

Additional transformations can include horizontal shifting (phase shift), which involves moving the graph left or right, and vertical shifting, involving moving it up or down. While the current example doesn't include these, becoming familiar with transformations enables students to complete more advanced problems.