### Video Transcript

If ππ΄, which is equal to seven, 60 degrees, is a position vector in polar form of the point π΄ relative to the origin π, find the π₯π¦-coordinates of π΄.

First, letβs recall what it means for position vector to be given in polar form. The polar form of a vector π, π means that π is the magnitude of the vector and π is the angle that the vector makes with the positive π₯-axis. As the angle made by the vector ππ΄ with the positive π₯-axis is 60 degrees, this means that the vector is situated in the first quadrant. It, therefore, looks something like this.

Weβve been asked to find the π₯π¦-coordinates of the point π΄. And to do this, weβll consider sketching in a right-angled triangle below the vector ππ΄. In this right-angled triangle, we know one of the other angles, 60 degrees, and we know the length of one of the sides. In fact, itβs the hypotenuse of the triangle. ππ΄ is seven units.

The other two sides of the triangle give the values of π₯ and π¦. The horizontal side of the triangle will give the π₯-coordinate of point π΄. And the vertical side will give the π¦-coordinate of point π΄. As the triangle is right-angled, we can apply trigonometry in order to calculate π₯ and π¦.

Iβll begin by labelling the three sides of the triangle in relation to the angle of 60 degrees. π¦ is the opposite side of the triangle. π₯ is the adjacent. And seven is the hypotenuse. Now letβs recall the definition of two of the trigonometric ratios in a right-angled triangle.

Firstly, we know that the sine ratio in a right-angled triangle, sin of π, is equal to the opposite divided by the hypotenuse. Substituting the values for this triangle, this means that sin of 60 degrees is equal to π¦ over seven. And so we have an equation that we can solve in order to find the value of π¦.

First, we need to multiply both sides of the equation by seven. Now Iβve swapped the two sides of the equation round here. But we have that π¦ is equal to seven sin 60 degrees. And weβll come back to this in a moment.

Another trigonometric ratio in right-angled triangles is the cosine ratio. Cos of π is equal to the adjacent divided by the hypotenuse. Using the values for this triangle, this means that cos of 60 degrees is equal to π₯ over seven. And we have an equation that we can solve for π₯ in much the same way as we did for π¦. We need to multiply both sides by seven. So we have that π₯ is equal to seven cos 60 degrees.

Now we donβt need a calculator to answer this question because 60 degrees is a special angle for which the trigonometric ratios can be expressed exactly in terms of surds. And we need to remember what they are. Hereβs a reminder. Sin of 60 degrees is exactly equal to the square root of three over two. And cos of 60 degrees is exactly equal to the simple fraction one-half.

We can substitute the values for sin of 60 degrees and cos of 60 degrees into the expressions for π¦ and π₯. π¦ is equal to seven multiplied by root three over two. And π₯ is equal to seven multiplied by one-half. If I now write the π₯ and π¦ values as a pair of coordinates, so thatβs π₯ first and π¦ second, then we have that the π₯π¦-coordinates of the point π΄ are seven over two, seven root three over two.