# Trigonometric ratios of special angles

So far from lesson #1 to lesson #6, we computed the trigonometric ratios of special angles.

We summarize the results here before we move on to more complicated topics.

By special angles, we mean 30 degrees, 45 degrees, and 60 degrees.

These are the angles from the 45-45-90 degrees triangle and the 30-60-90 degrees triangle

In lesson #1, we saw that tan(45 degrees) = 1

In lesson #2 and lesson #4, we saw that sin(45 degrees) = 1 / √(2) = √(2) / 2

We also saw that cos(45 degrees) = 1 / √(2) = √(2) / 2

If you are not convinced that 1 / √(2) = √(2) / 2, use a calculator to prove this to yourself.

The reason we are using √(2) / 2 instead of 1 / √(2) might be because mathematicians prefer not to have the square root sign in the denominator.

In lesson #5 and lesson #6, we go the following results

cos(30 degrees) = y / 1 = y = √3 / 2

cos(60 degrees) = x / 1 = x = 1 / 2

sin(60 degrees) = y / 1 = y = √3 / 2

sin(30 degrees) = x / 1 = x = 1 / 2

The only thing we don't have are tan(30 degrees) and tan(60 degrees)

To get these, we will use the same triangle we use in lesson #5 to find the sine and cosine of 30 degrees and 60 degrees

tan(60 degrees) =

tan(60 degrees) =
2√3 / 2
= √ 3

tan(30 degrees) =
2 / 2√3

tan(30 degrees) =
1 / √3

1 / √ (3) = √(3) / 3 (Just like 1 / √ (2) = √(2) / 2 )

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