# Thread: I Can't Figure Out An Equation and A Word Problem

1. ## I Can't Figure Out An Equation and A Word Problem

Hello everyone, I'm having a bit of trouble figuring out how to solve these two problems
1. Solve for x if 0<=x<2pi: 4sin2x-cosx=5

2. A plane is flying with an airspeed of 244 miles per hour with heading 272.7 degrees. The wind currents are running at a constant 45.7 milers per hour in the direction 262.6 degrees the ground speed and true course of the plane.

If anyone is able to explain step by step what you do that would be best because I want to know what I'm doing wrong. I can't figure either of these out. Thanks in advance!

2. ## Re: I Can't Figure Out An Equation and A Word Problem

Originally Posted by badams6161
Hello everyone, I'm having a bit of trouble figuring out how to solve these two problems
1. Solve for x if 0<=x<2pi: 4sin2x-cosx=5

2. A plane is flying with an airspeed of 244 miles per hour with heading 272.7 degrees. The wind currents are running at a constant 45.7 milers per hour in the direction 262.6 degrees the ground speed and true course of the plane.

If anyone is able to explain step by step what you do that would be best because I want to know what I'm doing wrong. I can't figure either of these out. Thanks in advance!
The trick in (1) is to write

$\sin^2(x) = 1-\cos^2(x)$

let $u = \cos(x)$

then solve the resulting quadratic equation in $u$

then $x=\arccos(u)$

and double check the solutions to ensure they are solutions to the original equation

in (2) just convert the plane and wind velocity to vectors and add them.

The direction of the resulting vector is the true course and it's magnitude is the ground speed.

3. ## Re: I Can't Figure Out An Equation and A Word Problem

2. A plane is flying with an airspeed of 244 miles per hour with heading 272.7 degrees. The wind currents are running at a constant 45.7 milers per hour in the direction 262.6 degrees the ground speed and true course of the plane.
Air vector + Wind vector = ground Track vector

see attached sketch (not to scale) ...

angle of the triangle between the head of the air vector and tail of the wind vector is $180-(272.7-262.6)$

use the law of cosines to obtain the magnitude of the track vector (ground speed)

use the law of sines to get the angle between the air and track vectors,then determine the direction of the aircraft over the ground.

4. ## Re: I Can't Figure Out An Equation and A Word Problem

Thank you! For the first one my teacher gave us the wrong problem for homework since the one i posted gives imaginary numbers. That's why I couldn't figure it out. Thank you though!