4. Determine the number of ordered pairs (a, b) of positive...

4. Determine the number of ordered pairs (a, b) of positive integers with 0 < a < b < 50 so that \(\angle POQ = 45^\circ\) where P has coordinates P(a, b), Q has coordinates Q(2b, 3a), and O is the origin. \\ [Hint: the formula \(\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\) could be useful.]

Transcript

00:01 In this video, we're looking at angles a and b in the fourth quadrant and with this given information, we are trying to select which of these identities come out to a correct answer.

00:09 With respect to the quadrants, we've got four quadrants here, quadrant 1, 2, 3, and 4.

00:13 Total angles will be equal to 360.

00:16 Each quadrant will be divided to 90 degrees.

00:19 So let's look at the information with respect to what we understand about the quadrants.

00:24 So the general rule of these quadrants are that in the first quadrant, all trigonometric identities are positive.

00:30 And in the second quadrant, we have the sign and co -seekin to be positive.

00:33 In the third quadrant, we have tan and cart to be positive.

00:35 And fourth quadrant, we have cause and second to be positive.

00:38 Anything apart from these positives within the quadrant is mentioned here, there would be negatives.

00:43 So considering that the fourth quadrant is spanning from 270 to 360 degrees, we will translate the given information with respect to what is given in the problem.

00:54 So, for instance, if you look at what is given, it says tan a equals to minus 3 by 4.

01:00 And cos b equals to 1 by 3.

01:01 So with respect to a and b we need to establish the technometric identities by using the pythagoras theorem.

01:07 It would be easier to doing that.

01:09 So for angle a, if you look at how the tan of minus 3 by 4 can be established, would be having the angle a and the tan is nothing but the opposite side by the adjacent side.

01:21 Therefore you would get minus 3 and minus 4 as a ratio.

01:24 So you would get minus 3 by 4 for an angle a.

01:27 And using the pythagoras theorem we can prove that the hypotenuse would be naturally 5 and with that we can prove that sign of a will be minus 3 by 5 and cos of a will be 4 by 5.

01:39 So that would that summarizes what the angle a and its trigonometric identities would look like in a right angle to angle.

01:46 Similarly with respect to angle b we can prove that with the given information cost b equals 1 by 3, you can establish what sign b would be and tan b would be.

01:54 So these are very useful if it is in hand prior to solving the now let's look at the problem.

02:01 The problem they have given, they've actually given us about one, two, three, four, five problems here...

Question

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