Screencast of exercise 3. Next page - Content - The sine and cosine rules. Content The four quadrants The coordinate axes divide the plane into four quadrants, labelled first , second , third and fourth as shown. We can get even more general: the six function values for any angle equal the function values for its reference angle, give or take a minus sign. Therefore the signs of sine and cosine are the same as the signs of y and x.
But you know which quadrants have positive or negative y and x , so you know which angles or numbers have positive or negative sines and cosines. And since the other functions are defined in terms of the sine and cosine, you also know where they are positive or negative. Spend a few minutes thinking about it, and draw some sketches. Therefore x is positive, and the cosine must be positive as well.
You can check your thinking against the chart that follows. Its purpose is to show you how to reason out the signs of the function values whenever you need them, not to make you waste storage space in your brain. What about other angles? You can analyze negative angles the same way. The techniques we worked out above can be generalized into a set of identities.
You know that one will be in Q I and the other in Q II, and you also know that one will be the reference angle of the other. Therefore you know at once that the sines of the two angles will be equal, and the cosines of the two will be numerically equal but have opposite signs.
Look at the triangle in Quadrant I. Since its hypotenuse is 1, its other two sides are cos A and sin A. The other three triangles are the same size as the first, so their sides must be the same length as the sides of the first triangle. The thin arcs near the center of the circle trace the rotations. All of them have a reference angle equal to A. The relations are summarized below.
The identities for tangent are easy to derive: just divide sine by cosine as usual. Two examples:. One final comment: Drawing pictures is helpful to avoid memorizing things, but you might not consider it a rigorous proof of equation But if you move all the way around a circle, in either direction, you end up where you started. But if you can go around the circle once, you can go around the circle any number of times. Their values repeat over and over again. Of course secant and cosecant, being reciprocals of cosine and sine, must have the same period.
Nearly there now! What about tangent and cotangent? To get the most benefit from these problems, work them without first looking at the solutions. Refer back to the chapter text if you need to refresh your memory. You may want to skip this section, especially the first time you read the chapter. In that case, if you find something hard to follow, you may want to come back here for another approach.
Almost every repetitive process is governed by sines and cosines. How can we modify a sine or cosine function to have any desired period? Well, suppose you want a process that repeats every second. To speed things up, multiply t by some factor greater than 1.
What is more, both fall in the same left-or-right half of the x - y plane. The sign of the cosine depends only on which half. The sign of the sine depends on which of those halves. Problem 5. Use the previous theorem to evaluate the following. Example 4. Explain why. For they will be vertical angles, which are equal. Problem 6. And so on. Example 5. What are the rectangular coordinates x , y of the endpoint of the radius? Problem 7. What are the coordinates of B? In the second quadrant, the cosine is negative and the sine is positive.
Problem 8.
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