1. Understanding Trigonometry Through a Right Triangle
Trigonometry begins with a right-angled triangle. When we choose one of the acute angles (say \(\theta\)), the three sides of the triangle get special names:
- Hypotenuse: the longest side, opposite the right angle.
- Opposite side: the side opposite to angle \(\theta\).
- Adjacent side: the side next to (touching) angle \(\theta\), but not the hypotenuse.
All trigonometric ratios are simply ratios of these sides. Nothing more.
2. Definition of the Six Trigonometric Ratios
Trigonometric ratios relate an acute angle \(\theta\) of a right triangle to the lengths of its sides. Here are the standard definitions:
:
\( \sin \theta = \dfrac{\text{Opposite Side}}{\text{Hypotenuse}} \)
\( \cos \theta = \dfrac{\text{Adjacent Side}}{\text{Hypotenuse}} \)
\( \tan \theta = \dfrac{\text{Opposite Side}}{\text{Adjacent Side}} \)
\( \cot \theta = \dfrac{\text{Adjacent Side}}{\text{Opposite Side}} \)
\( \sec \theta = \dfrac{\text{Hypotenuse}}{\text{Adjacent Side}} \)
\( \csc \theta = \dfrac{\text{Hypotenuse}}{\text{Opposite Side}} \)
2.1. Pairing of Ratios (Quick Memory Tip)
Certain ratios go together naturally:
- sin ↔ cosec (reciprocals)
- cos ↔ sec (reciprocals)
- tan ↔ cot (reciprocals)
This helps in remembering formulas quickly.
3. Visualising Ratios with a Simple Triangle
Take a right-angled triangle with sides:
\( \text{Opposite} = 3,\quad \text{Adjacent} = 4,\quad \text{Hypotenuse} = 5 \)
This is the classic 3–4–5 triangle. Using it:
- \(\sin \theta = \dfrac{3}{5}\)
- \(\cos \theta = \dfrac{4}{5}\)
- \(\tan \theta = \dfrac{3}{4}\)
- \(\cot \theta = \dfrac{4}{3}\)
- \(\sec \theta = \dfrac{5}{4}\)
- \(\csc \theta = \dfrac{5}{3}\)
3.1. Example (Using a Real Triangle)
If a ladder 10 m long reaches a window 8 m above the ground, the wall and the ground form a right triangle.
Here:
- Hypotenuse = 10
- Opposite side = 8
- Adjacent side = \(\sqrt{10^2 - 8^2} = 6\)
So, the trigonometric ratios for the angle between the ground and the ladder are:
\( \sin \theta = \dfrac{8}{10} = 0.8 \)
\( \cos \theta = \dfrac{6}{10} = 0.6 \)
\( \tan \theta = \dfrac{8}{6} = \dfrac{4}{3} \)
4. Relationship Among Ratios (Quick Recap)
Even before studying all identities, some basic relationships among ratios follow directly from their definitions. These are useful in converting one ratio into another.
4.1. Reciprocal Relations
\( \sin \theta = \dfrac{1}{\csc \theta} \)
\( \cos \theta = \dfrac{1}{\sec \theta} \)
\( \tan \theta = \dfrac{1}{\cot \theta} \)
4.2. Quotient Relations
\( \tan \theta = \dfrac{\sin \theta}{\cos \theta} \)
\( \cot \theta = \dfrac{\cos \theta}{\sin \theta} \)