+/- angles, coterminal angles and values of trig functions

Question

What are some tips or tricks you would give to someone who is just starting to learn about positive and negative angles, coterminal and reference angles, and the values of trig functions?

Answer 1

To Henry's excellent tips, I'd add a couple:

1)The unit circle begins at the ero point, at the extreme right (on the positive x-axis). Positive angles increase in the counterclockwise direction, negative angles go clockwise.

2) Coterminal angles are angles with different values, each of which lie at the same point on the unit circle. Since once around the is 360 degrees, Coterminal angles are usually 360 degrees apart, e.g. , 30 degrees and 390 degrees.

3) As Henry points out, the x and y side lengths are related in each of the four quadrants of the unit circle; they differ in SIGN. A reference angle refers to the difference between your angle and the nearest x-axis (positive or negative). The RA is always less than 90 degrees, and, by convention, positive. It allows you to find the absolute value of the relevant trig function (sin,cos or tan). The SIGN of the function is determine by a mnemonic ASTC (often = All Students Take Calculus). it tells us the sign of the functions in each quadrant:

A - Q1 - All functions are positive

S - Q2 - Only the sine is positive

T - Q3 - Only the tangent is positive

C - Q4 - Only the cosine is positive

4) Last, remember SOHCAHTOA, and that H in the Unit Circle is always = 1

Answer 2

Many students find themselves mystified when first learning about the unit circle. The values and measurements seem almost random. But this is not the case, of course.

To make a real breakthrough in understanding these trig concepts, I recommend going back to the side ratios of the special right triangles (30-60-90 and 45-45-90). Start by noting the exact relationship between these side lengths and the coordinates on the unit circle.

Second, review the positive and negative regions on the x-y coordinate plane. Here, too, you'll see a precise relationship between the positive and negative coordinates in the four quadrants and the positive and negative coordinates on the unit circle.

Finally, to help with remembering the radian measures, start by simply getting the 90, 180, and 270-degree measures memorized. Those are easy. All the other values are fractions of those. For instance, 90 degrees is π/2. Since 30° is 1/3 of 90°, then it must also be true that 30° is 1/3 of π/2 radians, which is π/6 radians.

Best wishes!