Lesson video

Hi, I'm Mr. Bond, and in this lesson, we're going to learn how to solve trigonometric equations involving the sine of x between zero and 360.

So here we have the graph of y is equal to sine x for values of x between zero and 360 degrees.

When y is equal to one, what is the value of x? Well, to help us visualise this, let's draw the line y is equal to one onto our graph.

So this will be a horizontal line like this.

We can see that the point at which y is equal to one intersects the graph of y is equal to sine x at this point.

So, when y is equal to one, x is equal to 90.

This is the same as being asked solve sine x is equal to one.

Here, again, we have the graph of y is equal to sine x for values of x between zero and 360 degrees.

This time, we need to find estimates for the solutions of sine x is equal to negative 0.

3.

Last time, we drew in the line y is equal to one.

Which line will we draw this time? We're going to draw the line y is equal to negative 0.

3.

That would go here.

So we can see that the two points this time at which the line y is equal to negative 0.

3 intersects the graph of y is equal to sine x are these to points here.

What are the x coordinates at these two points? X is approximately equal to 197, or x is approximately equal to 343.

Here's a question for you to try.

Pause the video to have a go and resume the video when you're finished.

Here's the solution to question number one.

So just like in our example, we're using the graph to estimate solutions.

So for part a, we would have needed to draw a horizontal line at y is equal to 0.

5 and then use that to estimate our two solutions and we'd do a similar thing for parts b, c, and d.

The printout that you might have of this question might mean that the graph is quite small.

So if you want to find more accurate estimates, it's better to use a more blown up version of the graph, so if you can get hold of one of those, that would be useful.

Here's a slightly different example, but again, we're going to use the graph of y is equal to sine x between zero and 360 degrees.

We may be asked to use this graph to help us solve equations of the form y is equal to the sine of axe, or y is equal to a sine x.

So, for example, use the graph to solve sine of three x is equal to 0.

5 between zero and 120 degrees.

Well, this is of the form y is equal to sine of axe.

We need to draw on a horizontal line, just like we have in our previous examples.

What are we going to draw? We're going to draw the line y is equal to 0.

5.

That'll be a horizontal line like this.

So we can see that the line y is equal to 0.

5 intersects the graph of y is equal to sine x at these two points.

So we can read off some approximations.

From the graph, we can see that when y is equal to 0.

5, x is equal to 30, or x is equal to 150.

Therefore, for sine three x is equal to 0.

5 when y is equal to 0.

5, three x is equal 30, or three x is equal to 150.

So therefore, x is equal to 10, or x is equal to 50.

Now we're going to solve an equation of the other form.

For example, use the graph to estimate the solution to three sine x is equal to 0.

3 for values of x between zero and 360.

This is of the form y is equal to a sine x.

Firstly, if three sine x is equal to 0.

3, then by dividing both sides of the equation by three, sine x is equal to 0.

1.

So we can draw the horizontal line onto our graph, y is equal to 0.

1.

And we can see that this intersects the graph of y is equal to sine x here and here.

So, from the graph, when y is equal to 0.

1, x is approximately equal to six, or x is approximately equal to 174.

Here are some questions for you to try.

Pause the video to have a go at the task and resume the video when you're finished.

Here's the solution to question number two.

As in our previous examples, for those equations of the form sine axe, so that's parts a, b, and c, we need to use the graph to find the values and then divide those values by two and five respectively.

In part d, we should divide both sides of the equation by two first and then draw a horizontal line in y is equal to negative 0.

3 before finding the estimates for our solutions.

That's all for this lesson.

Thanks for watching.