Exact Values from the Special Triangles

0:01Can we find sine, cosine and tangent of 30, 45 and 60 degrees exactly, without rounding?

0:09Draw a right triangle with two sides of 1 at the right angle. Its other two angles are both 45 degrees.

0:17By the Pythagorean theorem, the hypotenuse, the longest side, is the square root of 1 squared plus 1 squared: root 2.

0:27Sine is opposite over hypotenuse: 1 over root 2. Multiply top and bottom by root 2 to get root 2 over 2.

0:37Check: root 2 over 2 is about 0.7071, the same as a calculator gives for sine 45.

0:46Cosine, adjacent over hypotenuse, is also root 2 over 2. Tangent, opposite over adjacent, is 1 over 1, or 1.

0:57Now halve an equilateral triangle, with all sides 2 and all angles 60 degrees. Each half has angles of 30, 60 and 90 degrees.

1:08Its hypotenuse is 2 and its base is 1, so its height is the square root of 2 squared minus 1 squared: root 3.

1:18Careful: root 3 is opposite the 60 degree angle, not the 30. The 30 degree angle is opposite the shortest side, 1.

1:28From 30 degrees: sine is 1 over 2, cosine is root 3 over 2, and tangent is 1 over root 3, or root 3 over 3.

1:40From 60 degrees: sine is root 3 over 2, cosine is 1 over 2, and tangent is root 3 over 1, or root 3.

1:51Two triangles, with sides 1, 1, root 2 and 1, root 3, 2, give the exact values for 30, 45 and 60 degrees.