Using half angle formulas, find the exact value of tan of 15 degrees.
First of all, we need to remember the half angle formula of tangent. Tan of 𝜃 over two is equal to one minus cos 𝜃 over sin 𝜃. In our case, we have tan of 15 degrees. If we set 𝜃 over two equal to 15, then we can rewrite 15 degrees as 30 over two. Tan of 30 over two is equal to one minus cos of 𝜃 over sin 𝜃. And our 𝜃 value is 30. Cos of 30 degrees equals the square root of three over two. And sin of 30 degrees equals one-half.
We can substitute these values into our equation, the square root of three for cos of 30 degrees and one-half for sin of 30. We can rewrite the equation to say two over two instead of one minus the square root of three over two, divided by one-half. Combining these two terms, we have two minus the square root of three over two, divided by one-half. When we divide by a fraction, we multiply by its reciprocal. The two in the denominator and the two in the numerator cancel out. And we’re left with tan of 15 degrees equals two minus the square root of three over one.
The exact value of the tan of 15 degrees is two minus the square root of three.