1. Modeling with the Sine Function  Section 2 Exercises  3. Derivatives of Trigonometric Functions  Trigonometric Functions Main Page  "RealWorld" Page  Everything for Calculus  Español 
The two basic trigonometric functions are: sine (which we have already studied), and cosine. By taking ratios and reciprocals of these functions, we obtain four other functions, called tangent, secant, cosecant, and cotangent.
Cosine
Let us go back to the bicycle introduced in the preceding section, and recall that the sine of $t, \sin t,$ was defined as the $y$coordinate of a marker on the wheel. The cosine of $t,$ denoted by $\cos t,$ is defined in almost the same way, except that this time, we use the $x$coordinates of the marker on the wheel. (See the figure.)We often write this as
ant so we have found a relationship between the sine and cosine function.
Fundamental Trigonometric Identity

Let us now turn attention to the graph of the cosine function. The graph, as you might expect, is almost identical to that of the sine function, except for a "phase shift" (see the figure).
This gives the following new pair of identities.
Further Relationships Between Sine and Cosine
The cosine curve is obtained from the sine curve by shifting it to the left a distance of $\pi/2.$ Conversely, we can obtain the sine curve from the cosine curve by shifting it $\pi/2$ units to the right.
$\sin t = \cos(t  \pi/2)$ Alternative formulation We can also obtain the \cosine curve by first inverting the sine curve vertically (replace $t$ by $t$) and then shifting to the right a distance of $\pi/2.$ This gives us two alternative formulas (which are easier to remember)
$\sin t = \cos(\pi/2  t)$ 
Question
Since we can model the cosine function with a sine function, who needs the cosine function anyway?
Answer
Technically, that is correct; we don't need the cosine function and we can get by with the sine function by itself. On the other hand, it is convenient to have the cosine function around, since it starts at its highest point, rather than zero.
Modeling with the Cosine Function (General Cosine Curve)
Note that the basepoint is at the higher point of the curve. All the constants have the same meaning as for the general sine curve:
$C$ is the vertical offset (height of the baseline) $P$ is the period or wavelength (the length of each cycle) $ω$ is the angular frequency, given by $ω = 2\pi/P$ $α$ is the phase shift (the horizontal offset of the basepoint; where the curve reaches its maximum) 
* Source: Investment Company Institute/The New York Times, February 2, 1997. p. F8.
Solution
(a) Cosine modeling is similar to sine modeling: We are seeking a function of the formPutting this together gives
where t is time in years.
(b) To convert between a sine and cosine model, we can use the relations given above. Here, let us use the formula
Therefore,
$P(t) = 12.5\cos(0.157t) + 2.5$
$= 12.5\sin(0.157t + \pi/2) + 2.5.$
The Other Trigonometric Functions
As we said above, we can take ratios and reciprocals of sine and cosine to obtain four new functions. Here they are.
Tangent, Cotangent, Secant, and Cosecant

Solution
Since
we can enter this function as
$Y_1 = 1/\cos(x).$To set the window, let us use $2\pi ≤ x ≤ 2\pi,$ and $7 ≤ y ≤ 7.$ Here is the graph we obtain.
Question
What are the vertical lines doing here?
$\cos x = 0.$
If you have studied the section on limits in Chapter 3 of Calculus Applied to the Real World, or Chapter 10 of Finite Mathematics and Calculus Applied to the Real World, you will recognize this phenomenon in terms of limits; For instance,
Before we go on...
Here are the graphs of all four of these functions. You might try to reproduce them and think about the asymptotes
The Trig Functions as Ratios in a Right Triangle
Let us go back to the figure that defines the sine and cosine, but this time, let us think of these two quantities as lengths of sides of a right triangle:
We are also thinking of the quantity t as a measure of the angle shown rather than the length of an arc. Looking at the figure, we find that
$\sin t =$ length of side opposite the angle $t = \frac{\text{opposite}}{1} = \frac{\text{opposite}}{\text{\text{hypotenuse}}}$ 
$\cos t =$ length of side adjacent to the angle $t = \frac{\text{adjacent}}{1} = \frac{\text{adjacent}}{\text{\text{hypotenuse}}}$ 
$\tan t = \frac{\sin t}{\cos t} = \frac{\text{opposite}}{\text{adjacent}}$ 
This gives us the following six formulas
The Trigonometric Functions as Ratios in a Right Triangle
$\sin t = y$coordinate of point $P$  
$\cos t = x$coordinate of point $P$  
1. Modeling with the Sine Function  Section 2 Exercises  3. Derivatives of Trigonometric Functions  Trigonometric Functions Main Page  "RealWorld" Page  Everything for Calculus 
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