# 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(𝜃) = 0.8829 and 𝜃 is in the first quadrant, find cos(3𝜋/2 – 𝜃).

According to the identity developed before, cos(3𝜋/2 – 𝜃) = -sin𝜃. But what is sin(𝜃)? To find this, we need the Pythagorean identity, sin²(𝜃) + cos²(𝜃) = 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 – 𝜃) = -sin𝜃 = -0.4696

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

According to the identity developed before, tan(3𝜋/2 + 𝜃) = -1/tan𝜃 = -cos𝜃/sin𝜃. Again, we need the Pythagorean identity to find cos𝜃:

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

Therefore, tan(3𝜋/2 + 𝜃) = -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 – 𝜃) = -cos𝜃. In our case 𝜃 = 𝜋/6 and cos(𝜋/6) = √3̅/2 so cos(5𝜋/6) = -√3̅/2.

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

4. If sin𝜃 = 4/5 and 𝜋/2 < 𝜃 < 𝜋, find the exact value of cos𝜃

The part 𝜋/2 < 𝜃 < 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, cos𝜃 = – 3/5.