how did this happen?? TRIGNOMETRY HELP?

Question

tan20 = cot 70
sec 22.3 = csc 67.7
sin 1 = cos 99
cos(4x) = sin (90-4x)
cot n = tan(90 - n)

the answers are the one on the rigth... how did they get these answers?? pls help it my exams tomorrow

Answer 1

This happens because, in degrees, we have sin(x) = cos(90 -x) and cos(x) = sin(90 - x) for every x. x and 90 - x are called complementary arcs or angles. This implies that tan(90 - x) = cotan(x) and sec(90 - x) = csc(x), since tan(x) = sin(x)/cos(x), sec(x) = 1/cos(x), csc(x) = 1/sin(x)

There's a mistake in the 3rd statement: sin 1 = sin (90 -1) = sin 89, not sin 99.

Answer 2

Look at a nice right triangle. You've got a right angle, of course, but the other two angles are complementary, adding to 90 degrees. Because of SOH,CAH,TOA, the sine of one angle is the cosine of the other. Do you see that?

For the other functions -- it's the same concept. You know that tan and cot are reciprocals, as are sec and csc. Therefore, you've always got an equality when their angles sum to 90 degrees.

that's it! :)

Answer 3

Most of these are complementary angles.

1. tan 20: Draw a right angle with one angle 20, the complementary angle 70. The same ratio of sides that gives tan 20 also gives cot 70.

However, sin1 = cos 89

Answer 4

the inverse function is equal to 90-(angle)...

correct [sin(1) = cos (89)]