U17 Trigonometric Ratios (I) - Essential Formulas

This section covers the fundamental definitions of the three primary trigonometric ratios—sine, cosine, and tangent—for acute angles in a right-angled triangle. Understanding these ratios is the cornerstone for solving problems involving lengths and angles in right-angled triangles.

1 Definitions in a Right-Angled Triangle

The Three Basic Ratios

For an acute angle $\theta$ in a right-angled triangle, the three trigonometric ratios are defined relative to the sides: opposite (opp), adjacent (adj), and hypotenuse (hyp).

$$ \sin \theta = \frac{\text{opp}}{\text{hyp}} \quad \cos \theta = \frac{\text{adj}}{\text{hyp}} \quad \tan \theta = \frac{\text{opp}}{\text{adj}} $$
adjacent opposite hypotenuse θ

2 The Mnemonic: SOH-CAH-TOA

Memory Aid

A common mnemonic to remember the ratio definitions is SOH-CAH-TOA.

$$ \text{SOH: } \sin = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \text{CAH: } \cos = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \text{TOA: } \tan = \frac{\text{Opposite}}{\text{Adjacent}} $$

3 Relationship between $\tan \theta$, $\sin \theta$, and $\cos \theta$

Key Identity

From the definitions, we can derive an important relationship. For any acute angle $\theta$ where $\cos \theta \neq 0$, the tangent ratio is the quotient of the sine and cosine ratios.

$$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$

4 Special Angles: $30^\circ$, $45^\circ$, $60^\circ$

Exact Trigonometric Values

The trigonometric ratios for these common angles have exact values that must be memorized. They are frequently used in DSE examinations.

$$ \begin{array}{c|ccc} \theta & \sin \theta & \cos \theta & \tan \theta \\ \hline 30^\circ & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{3}} \\[10pt] 45^\circ & \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} & 1 \\[10pt] 60^\circ & \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} & \sqrt{3} \end{array} $$

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now