BQ# 2 -5 Unit T Concept Intro

• BQ#2
• The period for sine, cosine and tangent and cotangent differ because in sine and cosine, the period is 2pi because 2pi is the distance it takes for the pattern to repeat all over again. Since for sine, the 1st and second quadrant are postive and the 3rd and 4th ones are negative, (+,+,-,-) and for cosine, the 1st and 4th quadrants are postive, the rest negative, (+,-,-,+), it takes the whole unit circle (2pi) to repeat. However, in tangent and cotangent, it only takes half the unit circle (pi) because in tangent and cotangent, the pattern is (+,-,+-), and since the 2nd pair is the same as the 1st, the second pair does not need to be repeated, therefore it only takes up half of the unit circle.
• Both sin and cosine have amplitutes of 1 because that is as far as they can go in the unit circle, since the unit circle has the radius of 1, sine and cosine can only go up to -1 - +1. The ratio for sine is y/r and the ratio for cosine is x/r. r will equal one since we're dealing with the unit circle. Dealing with anything over 1 or less than -1will result in an error. Other ratios will work because they are not over one and are able to be greater than 1 and less than -1.
• BQ#5
• Sine and cosine do not have asymptotes whereas the other ratios do because of several reasons. Because the ratios of sine, cosine, cosecant, secant, tangent, and cotangent differ, so does thier place in the unit circle. Sine and cosine do not have asymptotes because they are both divided by 1 when looking at the ratio. If r=1, then we can not end up with asymptotes because we only get asymptoes when we divide by 0, which is why we get asymptotes in cosecant, secant, tangent and cotanget.
•  BQ#3
• The graphs of sine and cosine differ from the other graphs. Sine starts off going up (positive) and down (negaitive). Cosine is the inverse of sine, which goes down (negative) and up (positive). The cosecant graph is drawn by using the sine graph, however the valleys are used and asymptotes are involved to correctly graph. The secant graph is drawn by using the cosine graph, however the valleys are used and asymptotes are involved to correctly graph. The tangent and cotangent are both alike because thier quadrants stay the same sign (positive/negative), so when it has a negative, the graph goes down and when it has a positive, the graph goes up tp make a curvey, vertical wave.
• BQ#4
• A "normal" tangent graph is uphill and a "normal" cotangent graph is downhill because of a couple reasons. First of all, both tangent and cotangent are both positive in the first quardrant, negative in the second, positive in the third and negative in the fouth. With that being said, back to BQ#5, tangent has asymptotes when the line crosses at 0. The asymptotes hold the line in different quadrants. So when going uphill, the asymptotes are between quadrants 2 and 3 (negative postitive) and so fourth. The same rule would apply to cotangent with asymptotes.

Reflection #1 - Unit Q

24) What does it actually mean to verify a trig identity? To verify a trig identity, it means that it is true and you must prove that it is true by solving through and making x=x. We can't "touch" or move anything on the right side. The left side just equal the right side and if it does, we have verified the trig function.

25)What tips and tricks have you found helpful? A couple tips I found helpful is converting everything to cosine and sin, and working from there. When converting everything to sin and cos, we have more options to figure it out. I also found it helpful to take out a GCF and that helps make the problem simpler.

26)Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you. My thought process when working out a trig function is: 1) I look at the equation and check if there is a GCF to factor out. 2) I look for an identity. 3) If there is an identity, I substitute the identity and hope something will cancel out. If things do cancel out, then I try to look for something simple like costheta, and try to substitute it for 1/sec. When doing that, I hope something else cancels out and if it does, I try to simplify the rest.