### Video Transcript

The diagram shows a vector ๐ that
has a magnitude of 22. The angle between the vector and
the ๐ฅ-axis is 36 degrees. Work out the horizontal component
of the vector. Give your answer to two significant
figures.

In the problem, weโre being asked
to solve for the ๐ฅ-component of our vector, or ๐ sub ๐ฅ. With the components drawn in on our
diagram, we can see that the triangle is a right triangle. And therefore, we can use
trigonometry to solve for our unknown side. Letโs recall that we can use
SOH-CAH-TOA. SOH, the sin of the angle is equal
to the opposite side of the triangle divided by the hypotenuse. CAH, the cos of the angle is equal
to the adjacent side of the triangle divided by the hypotenuse. And TOA, the tan of the angle is
equal to the opposite side of the triangle divided by the adjacent side of the
triangle.

Looking back at our diagram, we are
solving for the ๐ฅ-component of our vector, weโre given the angle ๐, and weโre
given the length of the hypotenuse. Which one of our trigonometric
functions would we choose, sine, cosine, or tangent? The best one to choose is cosine
because our vector component is adjacent to the angle. Now, we need to isolate the
adjacent side of the triangle, as thatโs what weโre solving for. To do this, we multiply both sides
by the hypotenuse. This will cancel out the hypotenuse
on the right side of the equation and leave the left side of the equation as the
hypotenuse times the cos of ๐.

We can now use the values from the
problem to substitute in for our variables. The hypotenuse had a magnitude of
22, ๐ was 36 degrees, and ๐ sub ๐ฅ represents the horizontal component of our
vector. When we multiply 22 by the cos of
36 degrees, we get 17.8. In the problem, we were instructed
to give our answers to two significant figures. Looking at our answer right now,
itโs given to three significant figures. Therefore, we can round up 17.8 to
18. The horizontal component of a
vector that has a magnitude of 22 and makes an angle with the ๐ฅ-axis of 36 degrees
is 18.