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Section 1A - Mathematical Reflections

In Section 1A, Mathematical Reflections, we reviewed some of the key questions that we have encountered thus far in Chapter 1. The following is a list of key concepts that that are useful in solving the problems in Section 1A.

1a. What angle measure corresponds to 9pi/2 radians?

Since we know the standard angles in degrees and their radian equivalents (e.g. 45 degrees = pi /4, 90 degrees = pi/2), we can see that 9pi/2 radians is the same angle measure as pi/2, only revolved around the unit circle nine times (hence **9**pi/2). Subsequently, since 9pi/2 equals pi/2. one of our standard angles, we know that 9pi/2 radians = 90 degrees.

1b. For what value of x, where 0≤ x <2pi, will both cos x = cos 9pi/2 and sin x = sin 9pi/2

Consider this problem with the facts that cosine refers to the x-coordinate on a unit circle, and sine refers to the y-coordinate on a unit circle. Since we know the cosine value at 9pi/2 (which is 0), and we know the sine value at 9pi/2 (which is 1), we can tell this will be true for **no values of x**, since cos 9pi/2 = 0, and sin 9pi/2 = 1. Those values are constant and will never equal each other (as far as I know).

Key concept: converting radians to degrees and vice versa

2a. On a unit circle, locate approximately the point with coordinates (cos 3, sin 3).

See 2b.

2b. Estimate the values of cos 3 and sin 3.

Since we know the coordinate on the unit circle at pi = (-1, 0), and 3 is close to 3.14, the point will be somewhere very close to these numbers, for instance, somewhere around (-0.96, 0.5) or so, but hey, this is an estimation anyway.

2c. Use a calculator to estimate the values of cos 3 and sin 3 to four decimal places.

Using our handy dandy calculator, we can find that cos 3 = -0.9899, and that sin 3 = 0.1411

Key concepts: understanding where pi is on the unit circle and estimating radian values based on that. Also, how to use a calculator.

3a. On a unit circle, identify the places where cosine and sine are equal.

With the knowledge of //special// triangles (i.e. 45-45-90 and 30-60-90), it is apparent that this is the case with the triangle where two of the side lengths are equal (i.e. the 45-45-90 triangle), because that makes the ratios between opposite over hypotenuse (sine) and adjacent over hypotenuse (cosine) equal. And what angle do you have to measure from for this to be the case? Of course the 45 degree angle! Therefore, the 45 degree angle equivalent on our unit circle (pi/4) is one place where cosine and sine are equal. You might think that this is also the case in Coordinate II and Coordinate IV, but that is not quite right, because in those coordinates, sine is positive will cosine is negative and sine is negative while cosine is positive, respectively. This is however true in Coordinate III, where both sine and cosine are negative.

3b. Find all angle measures x where 0≤x<2pi, such that cos x = sin x

This is the same as 3a., only translated into degrees, not radians. So as we established above (and in class with the radian / degree unit circle quiz), pi/4 = 45 degrees. That negative value of cosine and sine is 5pi/4, which translates to 225 degrees, because it is 45 degrees + a 180 degree rotation about the circle.

Key concept: translating between radians and degrees and vice versa

4. Using graphs of cosine and sine on the interval 0≤x<2pi, find all values of x such that cos x = sin x.

Just look at the graphs and find the intersections. The x values of the intersections (they should be exact values) are the x-values.

Key concepts: graphing functions, finding intersections

5. Find all solutions if 0≤x<2pi - sin x * cos x - 2sin x = sin x

Treating the functions as variables, you can use algebra to eventually get cos x = 3. In fact, there are no places on a unit circle where cos x = 3 (since cos max is 1). Therefore, there are no solutions of x in this problem.

Key concepts: Algebra, treating the functions as variables, knowing the difference between a problem with no solutions and a human error.

6. Where are the turning points of the cosine and sine functions?

This means at what coordinate does a graph change from being increasing to decreasing. This problem can be done using either the unit circle or a Cartesian coordinate graph. In either case, this change occurs when one function reaches its peak value (max y-value) and then decreases to its lowest value (minimum y-value), and then begins to decrease again. If you look at a graph of these functions, you will see that these points are the same on both the unit circle and the Cartesian graph. For the cosine function, the turning points occur at every interval of pi because those are the instances of the max and min value for cos x. For the sine function, this turn occurs at every interval of pi / 2, because those are the instances of the max and min value for sin x.

Key concepts: Knowing the relationships between unit circle and the Cartesian graphical representation of trig functions, knowing "increasing" and "decreasing" functions.

7. What is a radian?

Think of the unit circle. Now! What is a radian? In the unit circle, you have radius of 1, and so the circle's circumference is 2pi. One entire revolution around the circle is 2pi, which, not accidentally, is the ratio between the arc length (2pi as I set it up to be) and the radius (1). So, a radian is the ratio between the length of an arc and its radius.

Key concepts: Knowing the Unit Circle like the back of your hand, I suppose(?)

8. How can you use a graph of y = sin x to estimate solutions to the equation sin x = -0.6?

In this case, you can just find the intersections (see problem 4). So, graph both graphs and then find their intersections.

Key concept: Using a graphical intersection to solve an equation