How can I rather quickly come up with the values of sine, cosine, and tangent for all the special angles?

Question

There's a quick and easy way to construct a chart of these values using a simple process.

Answer 1

I would recommend that you take a look at the unit circle if you haven't already. There is a pattern to it as far as the signs in each quadrant as well as the fact that all the measures that have the same reference angles have the same sin and cos values. So you really need to learn only one quadrant and you can derive the rest. The unit circle only has sin and cos. But you can calculate tan by taking sin/cos.

Answer 2

Step 1: Draw a table with the five special angles across the top and sine, cosine, and tangent down the side:

0 π/6 π/4 π/3 π/2

sine

cosine

tangent

Step 2: from left to right at sine, enter 0 under 0, 1 under π/6, 2 under π/4, 3 under π/3 and 4 under π/2:

0 π/6 π/4 π/3 π/2

sine 0 1 2 3 4

cosine

tangent

Step 3: from left to right at cosine, enter 4 under 0, 3 under π/6, 2 under π/4, 3 under π/3 and 4 under π/2:

0 π/6 π/4 π/3 π/2

sine 0 1 2 3 4

cosine 4 3 2 1 0

tangent

Step 4: divide all values in the table by 4:

0 π/6 π/4 π/3 π/2

sine 0/4 1/4 2/4 3/4 4/4

cosine 4/4 3/4 2/4 1/4 0/4

tangent

Step 5: take √ of each value in the table:

0 π/6 π/4 π/3 π/2

sine √0/√4 √1/√4 √2/√4 √3/√4 √4/√4

cosine √4/√4 √3/√4 √2/√4 √1/√4 √0/√2

tangent

Step 6: simplify:

0 π/6 π/4 π/3 π/2

sine 0 1/2 √2/2 √3/2 1

cos 1 √3/2 √2/2 1/2 0

tangent

Step 7: complete the tangent row by simplifying tangent=sine/cosine

0 π/6 π/4 π/3 π/2

sine 0 1/2 √2/2 √3/2 1

cos 1 √3/2 √2/2 1/2 0

tangent 0 1/√3 1 √3 ∞

Done!