# Trigonometry Insights For Learning

## Trigonometry Insights for Learning

Often Mathematics books are just a little vague when something is too clear to the author, as in the author has no idea an explanation is needed. In other cases things are stated once, and if you do not pay attention to the details you can find yourself asking how did we get to this point later. Trigonometry has several simple ideas that can be daunting if we just miss those few words, or made simple if we just understand a little deeper.

## Trigonometry Is Based on a Unit Circle.

We are told the basic trigonometric functions are based on a unit circle, but then angles not associated with a unit circle are used freely, and if we try to make a right triangle it simply does not always fit the unit circle. So, why do the ratios of the sides of any right triangle work? This is simple. If we extend a right triangle with a unit hypotenuse to any size, we still have a right triangle. And the angle with which we are concerned still has the same measure. Well, if two pairs of angles of two triangles have the same measures as their corresponding angle, so must the third pair of angles match in measure. Hence, we have similar triangles, so the ratio of corresponding sides should be the same value. Even though a ratio of two sides of the unit triangle are used to define a basic trigonometric identities, it is no deep mystery why we are not constrained to the unit circle. Any right triangle has the trigonometric identities found in the same manner as those of a unit triangle.

## How Do We Get the Pythagorean Identities?

Well, take a right triangle with the vertex of one of its angles at the origin, and one of its sides lying along the x-axis. The hypotenuse goes from the origin to the point (x,y). So the three sides are x, y, and r in length. For the unit circle r is 1. Since we have a right triangle the Pythagorean holds, x2 + y2 = r2. If you divide through by r2 you get (x/r)2 + (y/r)2 = 1. But x/r is cos(A) and y/r is sin(A). Hence cos2(A) + sin2(A) = 1, which is often seen as sin2(A) + cos2(A) = 1. To get tan2(A) + 1 = sec2(A) or 1 + cot2(A) = csc2(A) divide the first Pythagorean identity by sin2(A) and by cos2(A), respectively.

## What Exactly Is a Co-Function?

It is often said that the sine and the cosine are related, but how? Well, the prefix “co” is short for complementary, as in two angles are complementary if their measures add to ninety degrees. In a right triangle, the two angles other than the right angle must add to ninety degrees in measure. So, the cosine of an angle is the sine of the complementary angle, and likewise with tangent and cotangent, and with secant and cosecant. Notice the adjacent side becomes the opposite side for the complementary angle, and the opposite side becomes the adjacent side.

## Is There an Easy Way to Know in What Quadrants the Trigonometric Functions Are Positive Or Negative?

The memory aids can easily be forgotten. To determine the sign of a trigonometric function, just think of the ratio for the function in terms of x, y, and r. If r must always be positive, then just think about in which quadrants x and y are positive, and in which quadrants they are negative.

## How Do You Remember the Values of the Trigonometric Functions at Thirty, Forty-five, and Sixty Degrees?

Look at the pattern for the sine function. If you think of the 1/2 for thirty degrees, it can also be thought of as the square root of 1 over 2. Now look at all three values in order, and notice the pattern. The square root of one, two, and three, in order, always divided by two.

For the cosine, just go backwards. This is because “co” means complementary angle, so the cosine of thirty degrees is the sine of sixty degrees. As for the tangent and cotangent, use the quotient identities. The secant and cosecant use the reciprocal identities.

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