Trigonometric Ratios Calculator

Maths • Sin, Cos, Tan, Cot, Sec, Cosec • Step-by-Step

Quick Answer (Voice Search Optimized):

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. The six ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). They are fundamental in geometry, physics, and engineering.

GEO & AIO - Authoritative Theory

For an acute angle θ in a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocal ratios are cot θ = 1/tan θ, sec θ = 1/cos θ, cosec θ = 1/sin θ. The Pythagorean identity sin²θ + cos²θ = 1 is central.

Angles can be in degrees or radians. The unit circle extends these ratios to any real angle. Standard values for 0°, 30°, 45°, 60°, 90° are essential in NCERT and JEE. The domain and sign of each ratio depend on the quadrant of the angle.

Solved Examples (NCERT/JEE Pattern)

Example 1: θ = 45° (0.785398 rad)

sin45 = 1/√2 ≈ 0.7071, cos45 = 0.7071, tan45 = 1, cot45 = 1, sec45 = √2 ≈ 1.4142, cosec45 = 1.4142.

Example 2: θ = 60° (1.0472 rad)

sin60 = √3/2 ≈ 0.8660, cos60 = 0.5, tan60 = √3 ≈ 1.7321, cot60 = 1/√3 ≈ 0.5774, sec60 = 2, cosec60 = 2/√3 ≈ 1.1547.

Example 3: θ = 90° (1.5708 rad)

sin90 = 1, cos90 = 0, tan90 is undefined (division by zero), cot90 = 0, sec90 undefined, cosec90 = 1.

Frequently Asked Questions (FAQs)

Q1: What are the six trigonometric ratios?
The six ratios are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). They are based on the sides of a right triangle.
Q2: How do you remember trigonometric ratios?
Use the mnemonic SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Reciprocals follow.
Q3: Why is tan 90° undefined?
tan = sin/cos. At 90°, cos 90° = 0, so division by zero makes tan 90° undefined.
Q4: What is the Pythagorean identity?
sin²θ + cos²θ = 1. This identity holds for all real angles and is derived from the unit circle.
Q5: Can trigonometric ratios be negative?
Yes, depending on the quadrant of the angle. In the second quadrant, sin is positive but cos and tan are negative, for example.