Trigonometry Basics: Sin, Cos and Tan Explained
Trigonometry studies the relationship between the angles and sides of a triangle. For CBSE and ICSE students in Class 10, it begins with three simple ratios in a right-angled triangle: sine (sin), cosine (cos) and tangent (tan). Once you understand these three ratios, most board exam and JEE trigonometry questions become far easier. This guide explains them from scratch.
The three sides of a right triangle
Pick one acute angle in a right triangle and call it θ (theta). Relative to that angle, the three sides are:
- Hypotenuse: the longest side, always opposite the 90° angle.
- Opposite: the side directly across from angle θ.
- Adjacent: the side next to θ that is not the hypotenuse.
If you ever need to find the hypotenuse from the other two sides, you will use the Pythagoras theorem.
SOH-CAH-TOA: the three ratios
The famous memory aid SOH-CAH-TOA defines the ratios:
- SOH: sin θ = Opposite ÷ Hypotenuse.
- CAH: cos θ = Adjacent ÷ Hypotenuse.
- TOA: tan θ = Opposite ÷ Adjacent.
A useful link is tan θ = sin θ ÷ cos θ. The other three ratios are just reciprocals: cosec θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.
Standard angle values
You must memorise the values for the standard angles. They appear in almost every board paper.
- sin: 0° = 0, 30° = 1/2, 45° = 1/√2, 60° = √3/2, 90° = 1.
- cos: 0° = 1, 30° = √3/2, 45° = 1/√2, 60° = 1/2, 90° = 0.
- tan: 0° = 0, 30° = 1/√3, 45° = 1, 60° = √3, 90° = undefined.
A neat trick: write 0, 1, 2, 3, 4 under sin, divide each by 4, then take the square root. That gives 0, 1/2, 1/√2, √3/2, 1 for sin from 0° to 90°. Reverse the order for cos. For more such shortcuts, see our fun maths tricks for students.
Worked example 1
A right triangle has the side opposite θ equal to 3 cm and the hypotenuse equal to 5 cm. Find sin θ, cos θ and tan θ.
The adjacent side = √(5² − 3²) = √(25 − 9) = √16 = 4 cm. So sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, and tan θ = 3/4 = 0.75.
Worked example 2
Evaluate sin 30° × cos 60° + cos 30° × sin 60°.
= (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4 = 1. This is actually sin(30° + 60°) = sin 90° = 1, which confirms the answer.
Mastering these basics builds the foundation for heights and distances in Class 10 Maths and the advanced identities you meet in JEE Maths.
Frequently asked questions
What does SOH-CAH-TOA stand for?
It is a memory aid: SOH means sin = Opposite/Hypotenuse, CAH means cos = Adjacent/Hypotenuse, and TOA means tan = Opposite/Adjacent. These are the three basic trigonometric ratios of a right-angled triangle.
Why is tan 90 degrees undefined?
Because tan θ = sin θ/cos θ, and cos 90° = 0. Dividing by zero is not defined in mathematics, so tan 90° has no finite value.
Do I need to memorise the standard angle table?
Yes. The values for 0°, 30°, 45°, 60° and 90° appear constantly in CBSE board exams. Use the "0,1,2,3,4 divided by 4 then square-rooted" trick to recall the sin row instantly.