Trigonometric Ratios of Some Specific Angles

1. Introduction: The angles you will meet again and again

You already know what sine, cosine, and tangent of an angle mean. They are just ratios of two sides of a right-angled triangle. A right-angled triangle is a triangle with one $90°$ angle.

Quick reminder of the names. For an acute angle $\theta$ inside a right triangle:

• "opposite" — the side facing the angle $\theta$.
• "adjacent" — the side next to $\theta$ that is not the longest side.
• "hypotenuse" — the longest side, always opposite the $90°$ angle.

And the three ratios: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Here $\theta$ (theta) is just a name for the angle, the way we use $x$ for an unknown number.

Now here is the thing. In almost every problem you will solve this year, the angle is one of just five values: $0°$, $30°$, $45°$, $60°$, $90°$. These five keep coming back. So instead of measuring them again and again, you learn their exact ratio values once. After that, you just recall them.

Think of it like the multiplication tables. You did not re-derive $7 \times 8$ every time. You learned it once. We are going to do the same for these angles.

Stop scrolling. Before reading on, ask yourself: which angle in a right triangle is always the biggest? Try it in your head.

(The $90°$ angle. The hypotenuse sits opposite it because the biggest angle faces the biggest side.)

You can now name the three ratios and recall the five special angles we are about to fill in.