Some Equations Are Always True for Example

Math 1330, Chapter 8, Section 4

Trigonometric equations

Some equations are always true – for example, sin 2 x  cos 2 x  1. These equations are referred to as identities. For any value of x, the equation is true:

sin 2 39  cos 2 39  1.

Some equations are true only for certain values of the unknown; these equations are called conditional equations (true on the condition that x = a specific value or one of a specific set of values).

We’ve worked at developing a whole lot of identities – double angle formulas, Pythagorean identities, equations like sin ( t) = sin t. Now, we’re going to work with conditional equations.

 2 Is a solution to the equation sin x =  ?  2

What are all the solutions to this equation?

Start by looking at the sine graph and the line representing the LHS of the equation:

1 Now sometimes these conditional equations have a quadratic form.

Find all solutions to the following equation in the interval [  

2sin 2 x  5sin x  2

2 Solve cos 4x = 1 on the interval [0, 

Let’s think about this fairly carefully. I want to show you the answer graphically first:

and then I want to solve it with algebra:

So now our algebra agrees with our graph.

3 It won’t always be the case that you get familiar point pairs as solutions to your work. For the following equation, note that you can solve for sin x…but you get stuck trying to figure out x

1 sin 2 x   0 16

There’s an answer, of course: Here’s where the solutions are on the graph:

You can use your calculator: the sin 1 key and get a decimal approximation to x or you can leave the answer as

1 1 sin 1 ( )  x or sin 1 ( )  x 4 4

There are inverse function keys for the sine, cosine, and tangent functions. They are subject to the following understandings:

So we talk about sin 1 t,cos1 t,and tan 1 t.

Note that none of these is the reciprocal function!!!!

Also be aware that some older terminology still shows up: arcsine, arccos, arctan.

4 Now let’s talk about solving conditional equations with these inverse functions. Note that to solve for x, I will use the inverse function of both sides of my original equation.

Solve for x: sin x = 1/6

1 x = sin 1 = 9.5407 = ______in radians? 6

This answer is only from Quadrant 1 from my calculator…now I have to add in the Quadrant 2 and all multiples part of the answer: x =

It requires great care to work with inverse functions. You must keep in mind the limits of the programming of your calculator. cos x =  0.578 cos 1 cos x  cos 1  0.578  125.31

Note that this answer is in Quadrant 2 due to the domain restriction implicit and programmed into arccos. What do I need to do to get all solutions?

5 Let’s look at another conditional equation and see what we get for solutions:

4 tan2 x- 5tan t + 1 = 0

tan t = 1 tan t = .25 famous not so use the graph use the calculator

t = tan 1 0.25  14.04 

Now give the complete solutions over the whole domain of tan t:

6 Here’s another one: cos t  2  sec t

7 Here’s another one

1 1 cos t sin t  cos t  sin t  = 0 we’ll use factor by grouping on this one 2 2

8 And another kind of problem all together:

Solve for x on [0, 2

1 log (tan x) = - 3 2