Trigonometry has two lives in Indian exams. In Class 10, it's all about right triangles and height-distance problems. In Class 11 and JEE, it expands into identities, equations, and inverse functions. Both lives need attention, and both are high-scoring if you're systematic about it.
The Six Trigonometric Ratios
In a right-angled triangle with angle θ, we label the sides relative to that angle: opposite (perpendicular), adjacent (base), and hypotenuse.
Six Trigonometric Ratios
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent = sin θ / cos θ
cosec θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ = cos θ / sin θ
A quick way to remember sin, cos, tan: SOH-CAH-TOA (Some Old Horses, Canter And Have Thorough Oats — pick whatever works for you).
Standard Angle Values — The Table You Must Know
This table is non-negotiable. Every exam uses it. Memorise it or you'll be stuck at step one.
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | undefined |
| cosec | undefined | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | undefined |
| cot | undefined | √3 | 1 | 1/√3 | 0 |
Memory trick for sin: Think of it as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. Simplify each. For cos, read the sin row in reverse.
💡 Expert Tip
Notice that sin and cos are complementary: sin θ = cos(90° − θ). So sin 30° = cos 60°, sin 45° = cos 45°, and so on. This single relationship saves you from memorising both rows separately.
Pythagorean Identities
These three identities are the foundation of almost every proof and simplification problem.
Three Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
The second and third identities follow directly from the first — divide through by cos²θ or sin²θ respectively. So you only really need to remember one.
Why these matter: Most "prove that" questions in CBSE Class 10 and 11 require you to manipulate one side of an expression using these identities until it matches the other side. The skill is pattern recognition — spotting which identity applies.
Compound Angle Formulas
These are Class 11 and JEE territory. Essential for solving equations and proving identities.
Compound Angle Formulas
sin(A + B) = sin A cos B + cos A sin B
sin(A − B) = sin A cos B − cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
Double Angle and Half Angle Formulas
Set A = B in the compound angle formulas and you get the double angle formulas:
Double Angle Formulas
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
tan 2A = 2 tan A / (1 − tan²A)
The three forms of cos 2A are equally valid. In problems, choose the form that cancels with what's already in the expression.
Trigonometric Equations — General Solutions
This is where Class 11 and JEE students get caught. You must write the general solution, not just the principal value.
General Solutions
If sin θ = sin α, then θ = nπ + (−1)ⁿα, where n ∈ ℤ
If cos θ = cos α, then θ = 2nπ ± α, where n ∈ ℤ
If tan θ = tan α, then θ = nπ + α, where n ∈ ℤ
Example: Solve sin 2x = √3/2
sin 2x = √3/2 = sin 60° = sin π/3
General solution: 2x = nπ + (−1)ⁿ(π/3)
So x = nπ/2 + (−1)ⁿ(π/6), n ∈ ℤ
🎯 Exam Insider
In JEE, trigonometric equation questions often ask for solutions in a specific interval like [0, 2π]. Find the general solution first, then substitute values of n to find which solutions fall in the given range.
Height and Distance Problems
This is the Class 10 application of trigonometry. Real-world problems involving towers, poles, ships, and buildings.
Key terms:
- Angle of elevation: The angle measured upward from horizontal to the line of sight.
- Angle of depression: The angle measured downward from horizontal to the line of sight.
💡 Expert Tip
The angle of depression from point A to point B equals the angle of elevation from point B to point A. This follows from alternate interior angles with a horizontal line. Draw the diagram and you'll see it immediately.
Standard setup: A person stands d metres from the base of a tower. The angle of elevation to the top is θ. What is the height h?
tan θ = h / d → h = d × tan θ
That's the fundamental relation. Every height-distance problem comes back to this.
Solved Examples
Example 1 (Easy): Find sin 30° × cos 60° + sin 60° × cos 30°
Recognise the right-hand side: this is sin(30° + 60°) = sin 90° = 1. You don't even need the table.
Example 2 (Medium): Prove that (1 − sin θ)/(1 + sin θ) = (sec θ − tan θ)²
LHS: Multiply numerator and denominator by (1 + sin θ):
= (1 − sin θ)² / (1 − sin²θ) = (1 − sin θ)² / cos²θ
= [(1 − sin θ)/cos θ]² = [1/cos θ − sin θ/cos θ]² = (sec θ − tan θ)² = RHS ✓
Example 3 (Height & Distance): A tower stands vertically. From a point 30 m away on the ground, the angle of elevation to the top is 60°. Find the height.
tan 60° = h/30 √3 = h/30 h = 30√3 ≈ 51.96 m
Answer: Height = 30√3 m
Example 4 (Hard, JEE level): Solve 2cos²x − 3sin x = 0 for x ∈ [0, 2π]
Replace cos²x = 1 − sin²x: 2(1 − sin²x) − 3 sin x = 0 2 − 2sin²x − 3sin x = 0 2sin²x + 3sin x − 2 = 0
Let t = sin x: 2t² + 3t − 2 = 0 (2t − 1)(t + 2) = 0 t = 1/2 or t = −2
Since −1 ≤ sin x ≤ 1, we discard t = −2. sin x = 1/2 → x = π/6 or x = 5π/6
Answers: x = π/6 and x = 5π/6
Exam-Specific Tips
CBSE Class 10:
- Height and distance problems carry 3-4 marks. Always draw a clear diagram. If two angles or two observers are involved, draw both triangles on the same base.
- "Prove that" questions using identities carry 3 marks. Work on one side only unless instructed otherwise.
- Standard angle substitution questions are 1-mark easy marks — don't drop those.
CBSE Class 11/12:
- The general solution formula must be stated explicitly. Writing only the principal value costs you marks.
- Allied angle transformations (90° ± θ, 180° ± θ, 270° ± θ) are tested frequently.
JEE Main:
- Multiple correct type questions often involve checking identities for specific values.
- Trigonometric equations asking for the number of solutions in a given interval require careful graphical thinking.
- NEET has basic trig in physics problems (forces, projectiles) — the standard angle values are essential there too.
5 Common Mistakes
⚠️ Common Mistake
Mistake 1: Mixing up sin⁻¹x and 1/sin x sin⁻¹x (arcsin) is the inverse function, not the reciprocal. The reciprocal of sin x is cosec x. These are completely different.
⚠️ Common Mistake
Mistake 2: Forgetting the general solution In Class 11 and JEE, writing only x = π/6 when sin x = 1/2 is incomplete. You lose marks. Always write the general solution unless a specific interval is given.
⚠️ Common Mistake
Mistake 3: tan 90° exists tan 90° is undefined, not infinity (for the purposes of writing). In MCQs, "tan 90° = ∞" is not a valid algebraic step.
⚠️ Common Mistake
Mistake 4: Angle of depression ≠ angle below horizontal in the triangle Some students draw the angle of depression as an angle inside the triangle at the base. The angle of depression is measured from the horizontal at the top point, but the angle in your right triangle is the same value — use alternate angles to justify this properly.
⚠️ Common Mistake
Mistake 5: Working on both sides simultaneously in a proof To prove LHS = RHS, transform one side to match the other. Writing "LHS = ... = RHS = ... = LHS" in circular logic scores zero.
Practice Questions
Q1. Find the value of: sin 45° + cos 45°
Q2. Prove: tan θ + cot θ = sec θ × cosec θ
Q3. From the top of a cliff 50 m high, the angle of depression of a boat is 30°. Find the distance of the boat from the foot of the cliff.
Q4. Simplify: (sin 30° + tan 45°) / (cosec 60° + sec 45°)
Q5. Find the general solution of: cos 2θ = 1/2
Q6. If sin θ = 3/5, find the value of tan θ (given θ is in the first quadrant).
Q7. Two poles of equal height stand on either side of a road 80 m wide. From a point between them, the angles of elevation are 60° and 30°. Find the height and the position of the point.
Q8. Solve: 2sin²x + sin x − 1 = 0 for x ∈ [0, 2π]
Frequently Asked Questions
What is the unit circle and do I need it for CBSE? The unit circle is a circle of radius 1. It defines trig ratios for all angles, not just 0° to 90°. For CBSE Class 10, you don't need it. For Class 11 and JEE, it's essential — signs of ratios in different quadrants come directly from it.
How do I remember which ratio is positive in which quadrant? The mnemonic "All Students Take Classes" (ASTC): All positive in Q1, Sine in Q2, Tan in Q3, Cos in Q4.
Are height and distance questions always right triangles? In Class 10, yes. In JEE/advanced problems, you might get oblique triangles requiring the sine rule or cosine rule. But CBSE 10 sticks to right-angled setups.
How many identities should I memorise? For Class 10: the three Pythagorean identities. For Class 11/JEE: additionally, compound angle, double angle, and sum-to-product formulas. Don't just memorise — derive them a few times so you can reconstruct them under pressure.
What's the difference between trigonometric identities and equations? Identities are true for ALL values of the angle. Equations are true only for specific values, which you need to find.
Is trigonometry tested in NEET? Not directly as a maths topic, but standard angle values appear constantly in physics — projectile motion, force components, optical angles. Students who know their trig table cold have a clear edge.