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Rutuja

In: Computers and Technology

Submitted By rutujak
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The graphs of the sine and cosine functions Let's continue to use t as a variable angle. A good way for human beings to understand a function is to look at its graph. Let's start with the graph of sin t. Take the horizontal axis to be the t-axis (rather than the x-axis as usual), take the vertical axis to be the y-axis, and graph the equation y = sin t. It looks like this.

First, note that it is periodic of period 2. Geometrically, that means that if you take the curve and slide it 2 either left or right, then the curve falls back on itself. Second, note that the graph is within one unit of the t-axis. Not much else is obvious, except where it increases and decreases. For instance, sin t grows from 0 to /2 since the y-coordinate of the point B increases as the angle AOB increases from 0 to /2.

Next, let's look at the graph of cosine. Again, take the horizontal axis to be the t-axis, but now take the vertical axis to be the x-axis, and graph the equation x = cos t.

Note that it looks just like the graph of sin t except it's translated to the left by /2. That's because of the identity cos t = sin (/2 + t). Although we haven't come across this identity before, it easily follows from ones that we have seen: cos t = cos –t = sin (/2 – (–t)) = sin (/2 + t).
The graphs of the tangent and cotangent functions The graph of the tangent function has a vertical asymptote at x = /2. This is because the tangent approaches infinity as t approaches /2. (Actually, it approaches minus infinity as t approaches /2 from the right as you can see on the graph.

You can also see that tangent has period ; there are also vertical asymptotes every units to the left and right. Algebraically, this periodicity is expressed by tan (t + ) = tan t.

The graph for cotangent is very similar.

This similarity is simply because the cotangent of t is the tangent of the complementary angle – t.
The graphs of the secant and cosecant functions The secant is the reciprocal of the cosine, and as the cosine only takes values between –1 and 1, therefore the secant only takes values above 1 or below –1, as shown in the graph. Also secant has a period of 2.

As you would expect by now, the graph of the cosecant looks much like the graph of the secant

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