INVERSE TRIG IDENTITIES

Take away point of the Investigation The Y-value of one trig function is the X-value of the inverse Remember that:

Word Problem Time... The height of an ocean wave, h, as time, t, passes can be modeled by the below function, where height is in metres and time is in seconds. During the first 5 s that you watch the wave, when will the height of the wave be 12 m?

PART I of SOLUTION BUT... We are not done!

PART II of SOLUTION This gives you one solution... ...but is there more than one instance where in the first five seconds the wave will be the height of 12 m? CHECK OUT THE PERIOD!!! PERIOD = 2 seconds CHECK OUT THE PHASE SHIFT!!! PHASE SHIFT = 3 Draw the graph to confirm - So, what's the answer?

ANSWER 0.63s 1.37s 2.63s 3.37s 4.63s By drawing the graph you will see that this is a cosine function, and its wavelength starts at the crest (or maxima) of the wave. Because the H.T. (horizontal translation or phase shift is 3), the crest of the first wave starts at 3. Since cosine functions express periodic motion, the wave is continuous, both to the left and right of this point. This means that at every two seconds to the left and right of 3, a wave reaches its maxima. Therefore, the wave crests are at 1, 3, and 5 seconds. BUT! This is not the point on the graph equalling 12 metres because you had calculated that one of these point was at 3.37 seconds on the x-axis.

So... Since we know the wave motion is consistent, the peaks occur at 1, 3, and 5 seconds, and that at 3.37 seconds the wave reaches 12 metres, we can make the accurate prediction that at every 0.37 seconds before or after a maxima, the wave is at 12 metres