SOLUTION: sec(5π/3)*tan(5π/4)-cot(2π/3)*sin(-π/3) Evaluate each expression and simplyfy. Can someone please help me? Thanks.

Algebra ->  Trigonometry-basics -> SOLUTION: sec(5π/3)*tan(5π/4)-cot(2π/3)*sin(-π/3) Evaluate each expression and simplyfy. Can someone please help me? Thanks.       Log On


   

Question 1077833: sec(5π/3)*tan(5π/4)-cot(2π/3)*sin(-π/3)
Evaluate each expression and simplyfy.
Can someone please help me? Thanks.
Found 2 solutions by josmiceli, KMST:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Think of +5%2Api%2F3+ as +6%2Api%2F3+-+pi%2F3+
which is +2%2Api+-+pi%2F3+
This makes a negative 60 degree angle with
the 0 degree vector, so
+sec%28+5%2Api%2F3+%29+=+1%2Fcos%28+-pi%2F3+%29+
+1%2Fcos%28+-pi%2F3+%29+=+2+
----------------------------------
+tan%28+5%2Api%2F4+%29+=+tan%28+4%2Api%2F4+%2B+pi%2F4+%29+
This is a tangent in the 3rd quadrant which
is (-)/(-) and is positive
+tan%28+5%2Api%2F4+%29+=+1+
------------------------------
+cot%28+2%2Api%2F3+%29+=+cot%28+3%2Api%2F3+-+pi%2F3+%29+
This is a cotangent in the 2nd quadrant
which is negative
+cot%28+2%2Api%2F3+%29+=+-1%2Fsqrt%283%29+
------------------------------
+sin%28+-pi%2F3+%29+ is a sine function in the 4th quadrant
which is negative.
+sin%28+-pi%2F3+%29+=+-sqrt%283%29%2F2+
------------------------------------------
Putting it all together:
+2%2A1+-+%28+-1%2Fsqrt%283%29+%29%2A%28+-sqrt%283%29%2F2+%29+
+2+-+1%2F2+=+3%2F2+
---------------------
Check the math. I just used trig functions of
+pi%2F3+ in different quadrants
and also a function of +pi%2F4+ in the 3rd quadrant

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
1) If you know (or figure out, or look up)
the values of the main trigonometric functions
(cosine, sine, and tangent)
for pi%2F3 and pi%2F4 ,
you are 1/3 of the way to the answer.


2) You need to know some trigonometric identities that are really definitions:
cot%28theta%29=1%2Ftan%28theta%29 , and sec%28theta%29=1%2Fcos%28theta%29 .

3) You also need to understand the relation between
angles in other quadrants and
their symmetrical first quadrant reference angles.

Foe example (and relevant to the problem)
the diagram shows you that


There are diagrams, tables and charts that can help you.
Your textbook probably has then.
Your teacher probably handed out that kind of help.
You can find them online searching for "trig identities."
Wikipedia has a nice diagram, and a table.
However, you can figure out the answers by yourself,
with the Pythagorean theorem.
For pi%2F3 think of half of an equilateral triangle.

The Pythagorean theorem lets you figure out
the length of the long leg as h=sqrt%283%29%2F2 .
So, for the pi%2F3=60%5Eo , we have the trigonometric ratios
cos%28pi%2F3%29=1%2F2 , sin%28pi%2F3%29=sqrt%283%29%2F2 , and tan%28pi%2F6%29=%28sqrt%283%29%2F2%29%2F%281%2F2%29=sqrt%283%29 .
For pi%2F4 think of half of a square.