# Properties of the six trigonometric functions

## Inverse trigonometric functions

Three table look-ups, and computing a sum, a difference, and an average rather than one multiplication. First, recall that the domain of a function f x is the set of all numbers x for which the function is defined. It is convenient to have a summary of them for reference. We can see this in the graph, shown in Figure 5. No, not really. Note: This curve is still sinusoidal despite not being periodic, since the general shape is still that of a "sine wave'', albeit one with variable cycles. Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine. The Pythagorean formula for sines and cosines. Likewise, cot x, csc x, and sec x do not have an amplitude. You only need to know one, but be able to derive the other two from the Pythagorean formula. For example, tan x has neither a maximum nor a minimum, so its amplitude is undefined. This group of identities allow you to change a sum or difference of sines or cosines into a product of sines and cosines. The graph is shown in Figure 5. Note that there are three forms for the double angle formula for cosine. Sum, difference, and double angle formulas for tangent.

However, 3 sin x can never equal 3 for the same x that makes 4 cos x equal to 4 why? Note: This curve is still sinusoidal despite not being periodic, since the general shape is still that of a "sine wave'', albeit one with variable cycles.

Despite this, it appears that the function does have an amplitude, namely 2. This is a contradiction.

And the argument works for the other trigonometric functions as well. For example, tan x has neither a maximum nor a minimum, so its amplitude is undefined. They can all be derived from those above, but sometimes it takes a bit of work to do so.

We can see this in the graph, shown in Figure 5. To do this, we will use a proof by contradiction. Sum, difference, and double angle formulas for tangent.

### Graphs of trigonometric functions

And the argument works for the other trigonometric functions as well. Note: This curve is still sinusoidal despite not being periodic, since the general shape is still that of a "sine wave'', albeit one with variable cycles. There's not much to these. The Pythagorean formula for sines and cosines. Periodic functions with the same period and the same phase shift are in phase. We will now see how to shift the entire graph of a periodic curve horizontally. Thus, the amplitude is indeed 2. Three table look-ups, and computing a sum, a difference, and an average rather than one multiplication. Since the amplitude involves vertical distances, it has no effect on the period of a function, and vice versa. Despite this, it appears that the function does have an amplitude, namely 2. Defining relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine. In general, a combination of sines and cosines will have a period equal to the lowest common multiple of the periods of the sines and cosines being added.

The above may seem like a lot of work to prove something that was visually obvious from the graph and intuitively obvious by the "twice as fast'' idea.

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