How do I solve sin x x

Solve sine equations of the form sin (b x + c) + d = 0

Reading time: 6 min

Let's look at the zeros and note that we don't change the zeros when we stretch or compress the graph:

~ plot ~ sin (x); 2 * sin (x); 5 * sin (x); hide ~ plot ~

However, if we add up a value, all zeros change:

~ plot ~ sin (x) +0.5; 2 * sin (x) +0.5; 5 * sin (x) +0.5; 0.5; hide ~ plot ~

Each zero or each point of the zero shifts upwards.

Solution formula for zeros of sin (b x + c) + d = 0

Taking d into account, the following general solution formula for zeros can be derived:

Let's do another sample exercise to be more secure:

Solution formula:

Let's check the solution graphically:

~ plot ~ sin (2x + 30/180 * pi) -0.5; x = 0; hide ~ plot ~

We see the zero at.

If we need the solutions in the case of an unbounded interval, we still have to determine the period.

period
period

Period in radians

The zero formula is thus:

Let's draw the graph and see if we can find the zero again:

The first zero is at, another at. But there is also a second zero, how do we calculate this? To do this, we again use the identities:

However, our term is not x, but rather. We now have to use this for the identity formula:

Let's reshape that:

And let's now insert the zero.

Now we need to determine the x value that leads to.

The second zero is therefore at 60 °. Here, too, we define the period totals:

period
period

The solutions for the zeros summarized:

Tip: The program Zeroing for sine functions helps to check the correctness of the solutions that have been determined for various tasks.