Download 2.2.3 Trigonometric functions -‐ rules of calculation

2.2.3 Trigonometric functions -­‐ rules of calculation Welcome back! In the previous video we have introduced the trigonometric functions. From the definitions we can derive several important properties which lead to rules of calculation that we can use to solve equations involving these trigonometric functions. For instance, look at the rectangular triangle with hypotenuse equal to one. Then the Pythagorean Theorem leads to: cosine squared plus sine squared equals one. In fact, this holds for all real theta. From the unit circle, it is clear that both the cosine and the sine are periodic functions. In fact, if we make a complete turn counterclockwise, which is the positive direction, we run through the same values of the angle theta again. This implies that the cosine of theta plus 2pi equals the cosine of theta, and that the sine of theta plus 2pi equals the sine of theta. Moreover, if we make a complete turn clockwise, which is the negative direction, we also run through the same values of the angle theta again. This implies that the cosine of theta minus 2pi equals the cosine of theta, and that the sine of theta minus 2pi equals the sine of theta. This implies that the unit circle can be used to find the values of both the cosine and the sine for all values of the angle theta. Both functions are periodic with period 2pi. The periodicity of the cosine and the sine also becomes clear from their graphs. A shift of 2pi in both the positive and negative direction leads to the same values for both the cosine and the sine. The graph of the tangent shows that this function is periodic too. However, it has period pi instead of 2pi. Back to the rectangular triangle. Consider the indicated angle in the top corner, which equals pi over 2 minus theta. For this angle the opposite side and the adjacent side of the rectangle are reversed, which shows that the cosine of pi over 2 minus theta equals the sine of theta, and that sine of pi over 2 minus theta equals the cosine of theta. The periodicity of both functions shows that these formulas hold for all values of theta. Back to the unit circle and consider the angle theta. Then, reflection in the horizontal axis shows that the cosine is an even function, [click] which means that the cosine of minus theta equals the cosine of plus theta, and that the sine is an odd function, which means that the sine of minus theta equals minus the sine of theta. The graph of the cosine is line symmetric in the vertical axis, which shows that the cosine is even and the graph of the sine is point symmetric in the origin, which shows that the sine is odd. The graph of the cosine also shows that the cosine of pi minus x equals minus the cosine of x and the graph of the sine also shows that the sine of pi minus x equals the sine of x. Finally, we will derive the so-­‐called addition formula for both the sine and the cosine. This is a formula for the sum of two angles. We start with the rectangular triangle with hypotenuse equal to 1. Then the opposite side of the angle alpha equals the sine of alpha, while the adjacent side equals the cosine of alpha. Now we add an angle beta and create two extra rectangular triangles. Consider the lower one with hypotenuse equal to the cosine of alpha instead of 1. Then we easily obtain the lengths of both the opposite and the adjacent side of the angle beta involving this factor cosine of alpha.............. (pause) Then we consider the upper rectangular triangle with hypotenuse equal to the sine of alpha instead of 1. Similarly, we obtain the lengths of both the opposite and the adjacent side of the angle beta involving this factor sine of alpha. So we have this Now we create an extra rectangular triangle with one of the angles equal to the sum of alpha and beta. Note that the hypotenuse of this triangle is equal to 1. This implies that the sine of alpha plus beta is equal to the length of the red sides. This is the addition formula for the sine. For the cosine of alpha plus beta we must have the green side, which is the difference between the blue side below and the orange side above. This leads to the addition formula for the cosine. If we take both alpha and beta equal to x, these formulas lead to the double angle formulas for the sine and for the cosine. This proof only holds for acute angles alpha plus beta. However, both formulas hold for all values of alpha and beta. All formulas derived in this video can be useful in solving equations involving trigonometric functions. Good luck with the exercises!