If 𝑂𝐴, which is equal to seven, 60 degrees, is a position vector in polar form of the point 𝐴 relative to the origin 𝑂, find the 𝑥𝑦-coordinates of 𝐴.
First, let’s recall what it means for position vector to be given in polar form. The polar form of a vector 𝑟, 𝜃 means that 𝑟 is the magnitude of the vector and 𝜃 is the angle that the vector makes with the positive 𝑥-axis. As the angle made by the vector 𝑂𝐴 with the positive 𝑥-axis is 60 degrees, this means that the vector is situated in the first quadrant. It, therefore, looks something like this.
We’ve been asked to find the 𝑥𝑦-coordinates of the point 𝐴. And to do this, we’ll consider sketching in a right-angled triangle below the vector 𝑂𝐴. 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, it’s the hypotenuse of the triangle. 𝑂𝐴 is seven units.
The other two sides of the triangle give the values of 𝑥 and 𝑦. The horizontal side of the triangle will give the 𝑥-coordinate of point 𝐴. And the vertical side will give the 𝑦-coordinate of point 𝐴. As the triangle is right-angled, we can apply trigonometry in order to calculate 𝑥 and 𝑦.
I’ll begin by labelling the three sides of the triangle in relation to the angle of 60 degrees. 𝑦 is the opposite side of the triangle. 𝑥 is the adjacent. And seven is the hypotenuse. Now let’s 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 𝜃, 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 𝑦 over seven. And so we have an equation that we can solve in order to find the value of 𝑦.
First, we need to multiply both sides of the equation by seven. Now I’ve swapped the two sides of the equation round here. But we have that 𝑦 is equal to seven sin 60 degrees. And we’ll come back to this in a moment.
Another trigonometric ratio in right-angled triangles is the cosine ratio. Cos of 𝜃 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 𝑥 over seven. And we have an equation that we can solve for 𝑥 in much the same way as we did for 𝑦. We need to multiply both sides by seven. So we have that 𝑥 is equal to seven cos 60 degrees.
Now we don’t 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. Here’s 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 𝑦 and 𝑥. 𝑦 is equal to seven multiplied by root three over two. And 𝑥 is equal to seven multiplied by one-half. If I now write the 𝑥 and 𝑦 values as a pair of coordinates, so that’s 𝑥 first and 𝑦 second, then we have that the 𝑥𝑦-coordinates of the point 𝐴 are seven over two, seven root three over two.