Research the origin of each trigonometric function’s name. Then answer the follo

Question

Research the origin of each trigonometric function’s name. Then answer the following questions. Cite any source(s) you use.

(a) To what does the “co” in cosine, cotangent, and cosecant refer? Draw an arbitrary right triangle with one acute angle having measure and the complementary angle having measure . What does this “co” relationship say about and and sine and cosine? How about tangent and cotangent or secant and cosecant? Label the lengths of the sides of your arbitrary right triangle a, b, and c, and show that this “co” relationship holds for values of all three pairs of trigonometric functions using the appropriate ratios of the sides.

(b) The names arcsine, arccosine, and arctangent come from what relationship between angle, arc, and these (partial) functional inverses of sine, cosine, and tangent?

Explanation / Answer

The same as it does in English:
The prefix co- is a Latin word meaning ''with'' or ''together''.

The cosine "partners" with the sine. cos(90°- x) = sin(x)
The cosecant "partners" with the secant. csc(90°- x) = sec(x)
The cotangent "partners" with the tangent. cot(90°- x) = tan(x)

Any trigonometric function (f), therefore, always satisfies either of the following equations:

f(q) = a / b   OR   f(a / b) = q,

where q is the measure of a certain angle in the triangle, and a and b are the lengths of two specific sides.

b) Inverse trigonometric relations are the inverse of the six trigonometric functions sine, cosine, tangent, cosecant, secant, and tangent. The corresponding inverse trigonometric relations of these functions are arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent respectively. If a trigonometric function is used in an equation with two variables, the roles of the two variables will be reversed if the inverse trigonometric function is to have the same meaning. For instance x = sin y has exactly the same meaning as y = arcsin x or y = sin-1x.

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