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How to Graph Sine and Cosine on the ACT Math Test

Sine, cosine, and tangent are the three main trig identities you might remember from Math class. You’ll only see them on the toughest ACT Math Test questions, but getting them correct can really raise your score!

These graphs are usually graphed and expressed in degrees, but you may also see them expressed in radians. There are 2π radians in one circle. Each point on a circle corresponds to a certain number of radians. To convert degrees to radians, simply multiply by π/180.

Sine and cosine both have standard graphs that you need to memorize for the ACT Math Test. The standard equation for sine looks like this:  y = sin x. The “period” of the wave is how long it takes the curve to reach its beginning point again. The coefficient in front of “sin” (here 1), is called the amplitude. It effects how high and how low the wave reaches vertically. If that coefficient changes, then the height changes. For example, y = 5 sin x, would show a curve that reaches +5 on the y-axis and extends down to -5 on the y-axis.


Cosine also has a standard equation. It looks like: y = cos x. For the graph of cosine, notice how it begins at its highest y-value and descends, whereas sine begins at the origin. Cosine and sine have the same period of 2π. Questions involving trig graphs will likely require you to match given equations with graphs, or interpret the meaning of certain graphs, such as in a question like this: What is the smallest positive value for x where y = cos 2x ?

The difference between y = cos x and y = cos 2x is that the coefficient in front of x is halving the period, so it will now take just one pi to complete its cycle. The smallest x-value for cosine usually occurs at π/2. For the new graph, it will occur at π/4, which is ½ of π/2.

How will the graph of the function f(x) = 4sin x + 0.2 differ from the graph of f(x) = sin x? 

A) The graph’s period will be 4 times as much and the graph will shift 0.2 units down.

B) The graph’s period will be 4 times as much and the graph will shift 0.2 units up.

C) The graph’s amplitude will be 4 times as much and the graph will shift 0.2 units down.

D) The graph’s amplitude will be 4 times as much and the graph will shift 0.2 units up.

We know that the coefficient in front of sine changes the amplitude, so (A) and (B) can quickly be eliminated, since 4 multiplies the amplitude by 4. Just like a linear equation, adding to the end of an equation shifts a graph upwards. For example, the only difference between y = 8x and y = 8x + 7 is that the latter is 7 places higher on the y-axis. The answer is (C).

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