SOLUTION: please help me with the following questions 1. Determine the value of the following expression without using a calculator? tan210degrees x sin 240 degrees x cos 180 degrees divid

Algebra ->  Trigonometry-basics -> SOLUTION: please help me with the following questions 1. Determine the value of the following expression without using a calculator? tan210degrees x sin 240 degrees x cos 180 degrees divid      Log On


   

Question 582305: please help me with the following questions
1. Determine the value of the following expression without using a calculator?
tan210degrees x sin 240 degrees x cos 180 degrees divided by 405 degrees sin (-225degrees)
2. If tan 25degreees = a, determine in terms of a the value of:
tan205degrees - tan115degrees divided by tan245degrees + tan335degrees.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The trigonometric functions of angles in the second, third and fourth quadrants have the same absolute value as the trigonometric functions of a reference angle in the first quadrant. The reference angle is a reflection on one of the axes (x axis or y axis, or on the origin.
The trigonometric functions of 0%5Eo, 30%5Eo, 45%5Eo, 60%5Eo, and 90%5Eo, are all you need when they say "without using a calculator."
1. 210%5Eo=180%5Eo%2B30%5Eo so tan%28210%5Eo%29=tan%2830%5Eo%29=sqrt%283%29%2F3
240%5Eo=180%5Eo%2B60%5Eo so sin%28240%5Eo%29=-sin%2860%5Eo%29=-sqrt%283%29%2F2
cos%28180%5Eo%29=-1
I don't know what 405%5Eo is doing, all by itself in your problem. I expected to see some trigonometric function of 405%5Eo, not just the angle measure. In any case,
405%5Eo=360%5Eo%2B45%5Eo, so the trigonometric functions for 405%5Eo are the same as for 45%5Eo. (They are co-terminal angles).
-225%5Eo%2B180%5Eo=-45%5Eo, so sin%28-225%5Eo%29=-sin%28-45%5Eo%29=-%28-sin%2845%5Eo%29%29=sin%2845%5Eo%29=sqrt%282%29%2F2
2. If tan%2825%5Eo%29=a, since 65%5Eo%2B25%5Eo=90%5Eo, tan%2865%5Eo%29=1%2Ftan%2825%5Eo%29=1%2Fa.
From there you can find the tangents of angles that have 65%5Eo or 25%5Eo as reference angle.
205%5Eo=180%5Eo%2B25%5Eo, so tan%28205%5Eo%29=tan%2825%5Eo%29=a
115%5Eo=180%5Eo-65%5Eo, so tan%28115%5Eo%29=-tan%2865%5Eo%29=-1%2Fa
245%5Eo=180%5Eo%2B65%5Eo, so tan%28245%5Eo%29=tan%2865%5Eo%29=1%2Fa
335%5Eo=360%5Eo-25%5Eo, so tan%28335%5Eo%29=tan%28-25%5Eo%29=-tan%2825%5Eo%29=-a