Let $\displaystyle I=3\sqrt{2}\int_{0}^{x}{\frac{\sqrt{1+\cos{t}}}{178\cos{t}}}dt$. If 0<x<π and $\displaystyle \tan{I}=\frac{2}{\sqrt{3}}$, what is x?
Sine and cosine make up the trigonometric world. All other trigonometric functions can be made from these two basic functions. Sine
* Variable $k$ is an integer Domain and Range are only given for real (not complex) numbers
Relation to the Unit Circle Let a line through the origin, making an angle of Î¸ with the positive half of the xaxis, intersect the unit circle. The x and ycoordinates of this point of intersection are equal to cos Î¸ and sin Î¸, respectively. The point's distance from the origin is always 1. Unlike the definitions with the right or left triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. The following video show the relationship of sine on the unit circle and the graph that results: One More Thing On Sine Cosine
* Variable $k$ is an integer Domain and Range are only given for real (not complex) numbers
The cosine function is very much like the sine function because $\cos\left ( \theta \right )=\sin\left ( \frac{\pi}{2}\theta \right )$ Sine & Cosine Together As you can see, the unit circle can be broken down into sine and cosine.
Sine and cosine are a great team. The rest of the trigonometric functions can be written as complex sine and cosine functions, which you will see in the coming weeks. The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity which can be written as $\sin^2\left ( \theta \right )+\cos^2\left ( \theta \right )=1$ . We could go and show the various identities of sine and cosine functions, but we figured you can Google them or go to http://en.wikipedia.org/wiki/List_of_trigonometric_identities and see them. A medieval city has the shape of a square and is protected by walls with length 500 meters and height 15 meters. You are the commander of an attacking army and the closest you can get to the wall is 100 meters. Your plan is to set fire to the city by catapulting heated rocks over the wall with an initial speed of 80 meters per second. At what range of angles should you tell your men to set the catapult? Assume that you will be firing from exactly 100 meters from the wall and that you will be firing perpendicular to the wall. We also assume that the walls of the city are impervious to heated rocks.

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