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:

Angle, ???? | sin???? | cos???? | tan???? |

0 | 0 | 1 | 0 |

????/6, 30° | 1/2 | √3̅/2 | 1/√3̅ |

????/4, 45° | 1/√2̅ | 1/√2̅ | 1 |

????/3, 60° | √3̅/2 | 1/2 | √3̅ |

????/2, 90° | 1 | 0 | Undefined |

????, 180° | 0 | -1 | 0 |

3????/2, 270° | -1 | 0 | Undefined |

Now let’s do some examples:

- If cos(????) = 0.8829 and ???? is in the first quadrant, find cos(3????/2 – ????).

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

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

According to the identity developed before, tan(3????/2 + ????) = -1/tan???? = -cos????/sin????. Again, we need the Pythagorean identity to find cos????:

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

\end{equation*}\]

Therefore, tan(3????/2 + ????) = -0.9848/0.1736 = -5.6729.

3. Find the exact value of cos(5????/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 5????/6 in terms of a common angle. Notice that 6????/6 – ????/6 = 5????/6. That is 5????/6 = ???? – ????/6. We have a trig identity for this: cos(???? – ????) = -cos????. In our case ???? = ????/6 and cos(????/6) = √3̅/2 so cos(5????/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 sin???? = 4/5 and ????/2 < ???? < ????, find the exact value of cos????

The part ????/2 < ???? < ???? 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, cos???? = – 3/5.