Lesson video

Hi, I'm Mr. Bond.

And in this lesson, we're going to solve trigonometric equations involving cosine x between zero and 360.

Here's the graph of y is equal to the cosine of x for values of x between zero and 360.

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

It would be a horizontal line where all of the y coordinates are equal to one.

So it would look like this.

And we can see that this intersects the graph of y is equal to the cosine of x in two places, here and here.

So we can see that when y is equal to one, x is equal to zero, or x is equal to 360.

This is the same as being asked, solve the cosine of x is equal to one.

Here's another example.

We're starting again, with the graph of y is equal to the cosine of x between zero and 360.

This time, we're asked to find estimates for the solutions of the cosine of x is equal to negative 0.

3.

So instead of drawing a horizontal line at y is equal to one.

This time, we're going to draw a horizontal line at y is equal to negative 0.

3 like this.

Again, we can see that this horizontal line intersects the graph of y is equal to cos x at two points, here and here.

So looking at the graph, x is approximately equal to 105 or x is approximately equal to 255.

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.

Just like in our previous example, we need to draw horizontal lines at y is equal to 0.

5, negative 0.

5, 0.

9 and negative 0.

6 in order to find estimates for our solutions.

If you'd like to find more accurate estimates, it would be best to use as large a graph of y is equal to cosine x as possible.

This will mean your answers are more accurate.

Here's a slightly more complex example, we may be asked to use the graph of y is equal to the cosine of x between zero and 360 to solve equations of the form, y is equal to the cosine of x or y is equal to A multiplied by the cosine of x.

For example, we could be asked to use the graph to solve cosine of three x is equal to 0.

5, between zero and 120.

So this is of the form y is equal to the cosine of axe, where a is equal to three.

First, we need to draw a horizontal line at y is equal to 0.

5.

That would look like this.

And we can see that this line intersects the graph of y is equal to the cosine of x at these two points.

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

5, x is equal to 60, or x is equal to 300.

But this is the graph of y is equal to the cosine of x, we were asked to solve the cosine of 3x is equal to 0.

5.

So for cosine 3x is equal to 0.

5.

when y is equal to 0.

5, 3x is equal to 60, or 3x is equal to 300.

And therefore, x is equal to 20 or x is equal to 100.

Now we're going to solve an equation of the other form y is equal to acos x.

So we could be asked to use the graph to estimate the solution to three cos x is equal to 0.

3 between zero and 360.

So firstly, if three cos x is equal to 0.

3.

Then by dividing both sides of the equation by three, this gives that the cosine of x is equal to 0.

1.

So drawing a horizontal line at y is equal to 0.

1 gives us this, and we can see this intersects the graph of y is equal to the cosine of x at these two points.

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

1, x is equal to 85, or x is approximately equal to 275.

Here are the final questions for this lesson.

Pause the video to have a go at the question and resume the video once you're finished.

Here are the solutions to question number two.

So just like in our previous examples, when we're finding estimates for the solutions, whether it's in the form, y is equal to the cosine of axe, we need to use the graph to find our values, and then divide these values by two for parts a and b, and by five for part c.

In part d, we had an equation of the form y is equal to a cos x.

So we need to divide both sides of this equation by two initially, and then draw a horizontal line in at y is equal to negative 0.

3 in order to find estimates for our solutions.

That's all for this lesson.

Thanks for watching.