Find tan of 30 degrees.
There are many special angles that we need to learn the trig values of. In particular, we need to learn sin, cos, and tan of 30 degrees, 45 degrees, and 60 degrees. If we consider the right-angled triangle shown, we can use right-angled trigonometry, or SOHCAHTOA, to work out sin, cos, and tan of 30 and 60 degrees.
In this example of a 30-60-and-90-degree triangle, the hypotenuse will be twice the length of the shortest side. We can work out the length of the third side, in this case, root three, using Pythagoras’ theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side, or hypotenuse, of any right-angled triangle.
In our triangle, we have one squared plus 𝑥 squared is equal to two squared. One squared is equal to one. And two squared is equal to four. Subtracting one from both sides of this equation gives us 𝑥 squared is equal to three. Finally, square rooting both sides gives us a value of 𝑥 equal to root three. As previously stated, this means that the third side of the triangle is equal to root three.
We can now use the trigonometrical ratios to calculate tan 30. Our first step is to label the sides. The longest side is the hypotenuse. The side opposite the 30 degrees is the opposite. And the side next to, or adjacent, to the 30-degree angle and the 90-degree angle is the adjacent. Our tangent, or tan, ratio states that tan 𝜃 is equal to the opposite over the adjacent.
In this question, tan 30 is equal to one over root three. We have, therefore, shown that the value of tan of 30 degrees is equal to one over root three. The sin of 𝜃 is equal to the opposite over the hypotenuse and the cos, or cosine, of 𝜃 is equal to the adjacent over the hypotenuse. We can, therefore, also see that the sin of 30 degrees is equal to one-half and the cos of 30 degrees is equal to root three over two.
We could use the same method to calculate the sin, cos, and tan of 60 degrees. In doing this, we would need to take care that the opposite and adjacent would swap places. This is because the root three is opposite the 60 degrees and the one is adjacent, or next, to it.