How can I find the exact values of a negative trig function!?

Question

Hi! I have done these before and understand how to get exact values. But I have only worked with triangles with positive sides and missing sides that I can use the Pythagorean Theorem to use. So how would I find exact values given, sinθ= negative 1 divided by the square root of 5! And tanθ < 0? I also don’t know if the negative should be applied to the numerator or denominator or both? How can I have a negative side? Thank you!

Answer 1

First of all the coordinates of the point on the triangle has negative coordinates. But the side measures are always positive, per the distance formula...

Second, if there is only one negative, just move it to the top... NEVER PUT an extra negative sign

where it does not belong!!! If there is only one negative sign, it CANNOT be in BOTH!!!

Now for the problem:

sin T = -1/sqrt(5) = -sqrt(5)/5 <--- rationalizes the denominator....

Next you need to remember the mnemonic device: ASTC "All students take calculus" or

"Add sugar to coffee"

A: all trig functions are positive in quadrant 1

S: sine is positive in quadrant 2

T: tangent is positive in quadrant 3

C: cosine is positive in quadrant 4

Here, the sine and tangent are both negative, so that must be quadrant 4.

the inverse sine of sqrt(5)/5 is about 26.565+ degrees

so the angle in quadrant 4 is 360 - 26.565+ = 333.435-