# When solving trig equations, how do you find ALL the answers?

- 1 vote

When finding all the values of x in a certain range say 0<x<360 for example. I know how you get from the equation to sinx= and then x= and what not. but then where do you go from there.

I know it involves adding/subtracting 360/180 in some cases. but how do you know what to do in each case? does it vary depending on if it's a tan, sin or cos graph?

i feel like i'm missing out on some really obvious point here. but i really don't understand how sometimes there's only say 2 answers and other times there's many more!

Here's an example of a question if i'm not making any sense...

**Solve the equation sin2x=sin48 giving the values of x in the interval 0<x<360**

:-)

## 2 Answers

- 2 votes

There are many ways to do it, but the simplest, which i find, is drawing the graph out and visually seeing how many solutions there are. It may take more time but it really does pay off if there is a twist in the question.

Mr. Whittington's solution does work very well, but if you are like me i prefer spending a few extra's minutes sketching the graphs quickly, drawing the solution line across the graph, and having a perpendicular line going to the x-axis, this showing you where the solutions are. Then simply add/subtract the principle angle from the closest know x-axis value (90,180,270,360) You should know the shape of the graph anyways.

But make sure you do the appropriate adjustments to your range if you use this method, for example in sin(2x-15*) you would minus 15* from the range and multiply the range by two, then once you have you solutions you can simply divide by 2 and add 15*

In other words if it has a transformation you do the opposite to shift it back to the original graph, solve it, then transform the solutions.

Hope this helps.

- 0 votes

Well that means that 2x=48. Do you know your 4 quadrants? if not, look them up. sine is positive in the first and second quadrant. at the moment you have 2x so you find the values as 2x. so sine is positive in quadrants one and two. we know that 48 is in quadrant one. quadrant two ends at 180, so 180-48 = 132. you have 2x=48 and 132 so then x=24, and 66.

The quadrants can be remember by, All Silly Tom Cats. All positive(quadrant 1), only sine is positive(quadrant 2), only tan positive (quadrant 3) and only cosine positive (quadrant 4). each quadrant is 90 degrees.

Edited by Josh Whittington on Thu 10th May, 2012 @ 11:46