Expert: Steve Holleran - 4/7/2008

Question
QUESTION: I'm interested in trying to find the ingenious technique to solve the following trigonometry equation:

tan(x) + cos(x) = 3

Can you offer a suggestion...thank you for your time and energy.

ANSWER: Hi Mark,

I've played with this one for days now, and I cannot get an algebraic solution.  I got it to a polynomial in cos x, but found no rational solutions.

The only way I see to solve it is graphically, using either a graphing calculator or online grapher.

Steve

---------- FOLLOW-UP ----------

QUESTION: Thank you for your response and energy on this exercise. What was the polynomial in terms of cos(x) that you came up with? Did you use a combination of standard trigonometry identities?

Again, thank you for your continued assistance.

Mark

Answer
Hi Mark,

Here's my thinking:

tan x + cos x = 3

sinx / cos x + cos x = 3

Multiply everything by cos x:

sin x + cos^2 x = 3 cos x

sin x = 3 cos x - cos^2 x

Square both sides:

sin^2 x = 9 cos^2 x - 6 cos^3 x + cos^4 x

1 - cos^2 x = 9 cos^2 x - 6 cos^3 x + cos^4 x

    0 = cos^4 x - 6 cos^3 x + 10 cos^2 x - 1.

The only possible rational zeros are cos x = +/- 1
but after trying them through synthetic division, neither works.

Still don't have an algebraic way to solve.

Steve