Trigonometry: Complete Guide

What is Trigonometry?

• Derived from Greek words: “trigono” (triangle) + “metry” (measurement).
• Focuses on angles and side lengths of triangles.
• Used in geometry, physics, engineering, astronomy, and navigation.

Basic Trigonometric Ratios

For a right-angled triangle with angle θ:

Ratio	Formula	Example
Sine (sin)	Opposite / Hypotenuse	$\sin \theta =\frac{a}{c}$
Cosine (cos)	Adjacent / Hypotenuse	$\cos \theta =\frac{b}{c}$
Tangent (tan)	Opposite / Adjacent	$\tan \theta =\frac{a}{b}$
Cotangent (cot)	Adjacent / Opposite	$\cot \theta =\frac{b}{a}$
Secant (sec)	Hypotenuse / Adjacent	$\sec \theta =\frac{c}{b}$
Cosecant (csc)	Hypotenuse / Opposite	$\csc \theta =\frac{c}{a}$

Trigonometric Ratios of Standard Angles

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

Trigonometric Identities

• ${\sin }^2\theta +{\cos }^2\theta =1$
• $1+{\tan }^2\theta ={\sec }^2\theta$
• $1+{\cot }^2\theta ={\csc }^2\theta$

These identities are essential for solving equations and proofs.

Applications of Trigonometry

• Engineering: Designing bridges, buildings, and machines.
• Astronomy: Calculating distances between stars and planets.
• Navigation: Determining ship and aircraft positions.
• Physics: Wave motion, optics, and mechanics.
• Daily Life: Measuring heights of trees, towers, and buildings.

Common Mistakes

• Confusing sine and cosine ratios.
• Forgetting to convert degrees to radians.
• Misapplying trigonometric identities.

Conclusion

Trigonometry is a powerful mathematical tool that connects geometry with real-world applications. By mastering ratios, identities, and formulas, students can excel in academics and apply these concepts in science, engineering, and everyday problem-solving.