Introduction to Trigonometry | Maths Grade 10 CBSE

Play with it

The three ratios, live

Drag the angle θ. The triangle reshapes and sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj update from the side lengths.

sin · cos · tan of θdrag the angle

angle θ 35°

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The three ideas in this chapter

In a right triangle, label the sides relative to an acute angle θ: the side opposite θ, the side adjacent to θ, and the hypotenuse (opposite the right angle). The three main ratios are:

sin θ = opp/hyp cos θ = adj/hyp tan θ = opp/adj

Remember them as SOH · CAH · TOA. Their reciprocals are:

cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
Also tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.

Common mistake: mixing up "opposite" and "adjacent". They are named relative to the angle θ you're using — the adjacent of one acute angle is the opposite of the other.

These five angles appear in nearly every exam question, so memorise the table:

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	—

Memory trick

For sin of 0/30/45/60/90, write √0, √1, √2, √3, √4 each over 2: 0, 1/2, 1/√2, √3/2, 1. Reverse it for cos. tan = sin/cos.

Watch out: tan 90° is not defined (cos 90° = 0, and you can't divide by zero).

Three identities follow from Pythagoras and hold for every angle:

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Worked example · find cos θ from sin θ

If sin θ = 3/5, find cos θ (θ acute).

Use sin²θ + cos²θ = 1 → cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25.
cos θ = √(16/25) = 4/5 (positive, since θ is acute).
Check with a 3-4-5 triangle: adjacent 4, hyp 5 → cos = 4/5 ✓.

Common mistake: writing sin²θ to mean sin(θ²). It means (sin θ)² — square the value of sin θ, not the angle.

Why this matters

Where you'll actually use this

Trigonometry turns angles into numbers — which is how we measure unreachable heights, draw every sound wave, and let phones and robots know which way they're pointing.

Every wave is a sine

Spin a point around a circle and its height traces a sine wave. That single curve is sound, light, radio, and the AC electricity in your home. Engineers describe music, earthquakes and Wi-Fi with sin and cos — the ratios from this chapter.

sin θ as height

Measuring unreachable heights

Stand a known distance d from a tower and measure the angle of elevation θ to its top. Then the height is simply h = d · tan θ — no climbing. Surveyors, sailors and astronomers have measured mountains, stars and oceans this way for centuries.

tan θ for heights

🎵 Music & sound

Each musical note is a sine wave at a fixed frequency; synthesisers add sines together.

🛰️ GPS & navigation

Phones and ships resolve direction and position using sine and cosine of bearing angles.

🤖 Robotics & animation

To move a joint or a character's arm to an angle, software uses sin and cos of that angle.

🔒 More real-world applications

Music, GPS, robotics and more — each explained with a diagram. Free to unlock.

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Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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