Expert: Paul Klarreich - 9/2/2009

Question
Solve for the indicated variable (listed after the semicolon). Restrict all answers to the interval [0,2pi] or [0,360degrees]

1) cosx + 2secx = -3; x
2) 2sin^2(45degress - theta) = 1-sin2theta; theta


Answer
Questioner: Sue
Country: United States
Category: Advanced Math
Private: No
Subject: Precalculus
Question: Solve for the indicated variable (listed after the semicolon). Restrict all answers to the interval [0,2pi] or [0,360degrees]

1) cosx + 2secx = -3; x
2) 2sin^2(45degress - theta) = 1-sin2theta; theta
.......................................................
Hi, Sue,

Now that you are starting precalc, you realize you need all those identities your trig teacher told you to memorize, but your fellow students all said "I'll never need this, why?" and you believed THEM instead of the teacher.

Now you need them.  Take 5 minutes each day, right after your blood-pressure and cholesterol pills (oh, yes -- you young kids don't need that stuff) and you'll learn them.

Study all the identities -- reciprocal, quotient, Pythagorean, sum-and-difference, double-angle -- and the graphs of sine and cosine.

One more thing -- read some other answers on this site, so you get an idea of how to type math into the computer. (Not easy, I know.)

1)  cosx + 2secx = -3
          2
cos x + ------ = -3    << recip id.
       cos x

C^2 + 2 = -3C        << mult through

C^2 + 3C + 2 = 0

(C + 2)(C + 1) = 0

cos x = - 2,    cos x = -1

Now you will get  x = pi as the solution.  (I leave that to you.)

2sin^2(45d - t) = 1-sin(2t)

sin(45 - t) = sin 45 cos t - cos 45 sin t  << sum identity

sin(45 - t) = sqrt(2)/2 [cos t - sin t]

(sin(45 - t)^2 = 1/2 [cos^2 t + sin^2 t - 2 sin t cos t]  << squaring.

(sin(45 - t)^2 = 1/2 [1 - 2 sin t cos t]

(sin(45 - t)^2 = 1/2 [1 - sin(2t)]  << double-angle

......................

Put it together:

1/2 [1 - sin(2t)] = 1 - sin(2t)

1 - sin(2t) = 2 - 2 sin(2t)

0 = 1 - sin(2t)

sin(2t) = 0.

Now 2t = 0, pi, 2pi, 3pi,  and you can finish up.