Question Video: Evaluating the Tangent Function of an Angle given the Cosine and Sine Functions Mathematics • First Year of Secondary School

Video Transcript

Find tan 𝐴 given sin 𝐴 is equal to 0.5 and cos 𝐴 is equal to root three over two.

In this question, we’re given the value of the sin and cos of angle 𝐴. And we need to use these to find tan 𝐴. One way to do this is to recall that the tan of angle 𝜃 is equal to sin 𝜃 over cos 𝜃. This means that in our question tan 𝐴 is equal to sin 𝐴 over cos 𝐴. Rewriting 0.5 as one-half, we have tan 𝐴 is equal to one-half divided by root three over two. When dividing two fractions, we multiply the first fraction by the reciprocal of the second. And one way to remember this is using the acronym KCF. We keep the first fraction the same, we change the division to a multiplication, and we flip the second fraction.

We can now multiply the numerators and denominators separately, giving us tan 𝐴 is equal to two over two root three. Dividing the numerator and denominator by a common factor of two, this simplifies to one over root three. If sin 𝐴 is equal to 0.5 and cos 𝐴 is equal to root three over two, then tan 𝐴 is equal to one over root three.

It is worth noting that we could rationalize the denominator by multiplying the numerator and denominator by root three. The fraction one over root three can be rewritten as root three over three. It is also worth noting that the values in this question correspond to one of our special angles; that is 30 degrees or 𝜋 over six radians. We know that the sin of 30 degrees is one-half, the cos of 30 degrees is root three over two, and the tan of 30 degrees is equal to one over root three. Whilst it was not required to answer this question, we know that one possible value of angle 𝐴 is 30 degrees.