# Trigonometry, Part 4

I will now do some examples of using the trig identities covered in the previous posts. But before I do, I want to show you a table that gives the values of the main trig functions for common angles:

Now let’s do some examples:

1. If cos(π) = 0.8829 and π is in the first quadrant, find cos(3π/2 – π).

According to the identity developed before, cos(3π/2 β π) = -sinπ. But what is sin(π)? To find this, we need the Pythagorean identity, sinΒ²(π) + cosΒ²(π) = 1:

$\begin{equation*} \text{sin}^{2} \theta +\text{cos}^{2} \theta \ =1\ \ \ \Longrightarrow \ \ \ \text{sin}( \theta ) =\sqrt{1-\text{cos}^{2} \theta } \end{equation*}$ $\begin{equation*} \text{sin}( \theta ) =\sqrt{1-0.8829^{2}} =0.4696 \end{equation*}$

Therefore, cos(3π/2 β π) = -sinπ = -0.4696

2. If sin(π) = 0.1736, and π is in the first quadrant, find tan(3π/2 + π).

According to the identity developed before, tan(3π/2 + π) = -1/tanπ = -cosπ/sinπ. Again, we need the Pythagorean identity to find cosπ:

$\begin{equation*} \text{cos}( \theta ) =\sqrt{1-0.1736^{2}} =0.9848 \end{equation*}$

Therefore, tan(3π/2 + π) = -0.9848/0.1736 = -5.6729.

3. Find the exact value of cos(5π/6) (without a calculator).

The clues here are that I have been developing trig identities and I just gave you a table of exact values of trig functions for common angles. Perhaps we can equate 5π/6 in terms of a common angle. Notice that 6π/6 – π/6 = 5π/6. That is 5π/6 = π – π/6. We have a trig identity for this: cos(π β π) = -cosπ. In our case π = π/6 and cos(π/6) = β3Μ/2 so cos(5π/6) = -β3Μ/2.

By the way, you can find a decimal approximation to β3Μ/2, but this will just be an approximation no matter how many decimal places you include since β3Μ/2 is an irrational number and has non-repeating decimals. So the exact value is β3Μ/2 since we have agreed that β3Μ is the notation for the exact square root of 3.

4. If sinπ = 4/5 and π/2 < π < π, find the exact value of cosπ

The part π/2 < π < π means that the angle is in the second quadrant which means the cosine is negative. As before, we can find the cosine via the Pythagorean identity:

$\begin{equation*} \text{cos}( \theta ) =\sqrt{1-(4/5)^{2}} =3/5 \end{equation*}$

But as the angle is in the second quadrant, cosπ = – 3/5.