Introduction to Trigonometry
Trigonometry is one of the most important branches of mathematics, with applications ranging from physics and engineering to navigation and computer graphics. This guide will help you master the fundamental concepts through clear explanations and practice problems.
The Six Trigonometric Ratios
In a right-angled triangle with angle θ:
- <strong>sin θ</strong>: = Opposite / Hypotenuse
- <strong>cos θ</strong>: = Adjacent / Hypotenuse
- <strong>tan θ</strong>: = Opposite / Adjacent
- <strong>cosec θ</strong>: = Hypotenuse / Opposite = 1/sin θ
- <strong>sec θ</strong>: = Hypotenuse / Adjacent = 1/cos θ
- <strong>cot θ</strong>: = Adjacent / Opposite = 1/tan θ
Memory Trick: SOH CAH TOA
- <strong>S</strong>: in = **O**pposite / **H**ypotenuse
- <strong>C</strong>: os = **A**djacent / **H**ypotenuse
- <strong>T</strong>: an = **O**pposite / **A**djacent
Standard Angle Values
These values appear in almost every trigonometry problem:
- sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
Pro tip: Memorize the sin values (0, 1/2, √2/2, √3/2, 1) and use the complementary relationship sin θ = cos(90° − θ) to derive cos values.
Essential Trigonometric Identities
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Reciprocal Identities
- cosec θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
Quotient Identities
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Practice Problems with Solutions
Problem 1 If sin A = 3/5, find cos A and tan A.
Solution: Using sin²A + cos²A = 1: cos²A = 1 − 9/25 = 16/25, so cos A = 4/5. tan A = sin A / cos A = (3/5)/(4/5) = 3/4.
Problem 2 Prove that (1 + tan²A)/(1 + cot²A) = tan²A.
Solution: LHS = sec²A / cosec²A = (1/cos²A) / (1/sin²A) = sin²A/cos²A = tan²A = RHS.
Problem 3 If tan θ = 12/5, find sin θ and cos θ.
Solution: Hypotenuse = √(12² + 5²) = √(144 + 25) = √169 = 13. sin θ = 12/13, cos θ = 5/13.
Problem 4 Find the value of 2sin30° × cos30° .
Solution: 2 × (1/2) × (√3/2) = √3/2.
Problem 5 Simplify: sin²45° + cos²45°.
Solution: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1. (This confirms the Pythagorean identity.)
Applications of Trigonometry
Heights and Distances Trigonometry is used to find heights of buildings, mountains, and towers using angle of elevation.
Navigation Ships and aircraft use trigonometry for course plotting and position determination.
Physics Wave motion, oscillations, and AC circuits all rely heavily on trigonometric functions.
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