You are here:

Advanced Math/Trigonometric Identities

Advertisement

Expert: - 3/15/2009


Question
Evaluate the ff:
1.) tan2A, given sin A = 3/5 and A Є QII
2.) sec 2x, given tan x = 5/3 and sin x< 0
3.) cos 2x, given cot x = 3/5 and csc x < 0
4.) cot 2B, given sin B = 12/13 and cos B < 0
5.) sin 2x, given tan x = -2 and cos x > 0

Answer
1) To be in QII, we know that the adjacent side must be negative.
Since the far side is 3 units and the hypoteneuse is 5 units,
we know the base must be -√(5²-3²) = -√(25-9) = -√16 = 4 units.
This makes the cos(A) -4/5. What we need to know, though,
is the tan(A).  It is -3/4.

Apply this formula: tan(2A) = 2tan(A)/(1-tan²A).

2) For the tan() to be positive and the sin() to be negative,
the angle must be in QIII.  To find the sec(2x),
what I think is best is to note that sec(2x) = 1/cos(2x).
The identity for cos(2x) = 1 - 2sin²(x).

3) The cotangent is the adjacent/opposite.  
This means the adjacent is 3 and the hypoteneuse is 5.  

Since the csc(x) is 1/sin(x),
we know that the angle is the QIII or QIV.

Using this, the other leg is √(5²-3²) = √(25-9) = √16 = 4,
but we don't even need this.  We can use the formula
cos(2x) = 2cos²(x) - 1.

4) Note that the cot(2B) = 1 / tan(2B).
Also remember that tan(2B) is 2tan(B)/(1 - tan²(B)).
Note that if the hypoteneuse is 13 units
and the far side is 12 units,
then the near side is found by saying its √(13²-12²) =
√(169-144) = √25 = 5.

5) To find sin(2x), remember that its 2sin(x)cos(x).
Given that the tan() is negatvie and the cos() is positive,
the angle is in QIV.  Since the tan(x) = -2,
we know the far leg is -2 and the near leg is 1.
This would make the hypoteneuse have length = √(1²+2²) = √(1+4) = √5.
The sin(x) would be -2/√5 and the cos(x) would be 1/√5.

Advanced Math

All Answers

Answers by Expert:

Ask Experts

Volunteer

Scott A Wilson

Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

Experience

Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people nearby, friends with math questions, and several people every day on the PC, and you're probably make one more.

©2012 About.com, a part of The New York Times Company. All rights reserved.