Is it correct to say: Confusing 'opposite' and 'adjacent' sides of the angle?

Opposite = side facing the angle. Adjacent = side next to the angle that is NOT the hypotenuse. Hypotenuse is always across from the right angle.

Is it correct to say: Writing tan θ = cos θ / sin θ?

tan θ = SIN over COS. cot θ = cos over sin. Learn the mnemonic 'SOH-CAH-TOA'.

Is it correct to say: Applying identities beyond 0°–90° at Class 10 level?

Class 10 syllabus restricts θ to acute angles (0° < θ < 90°). Broader domain comes in Class 11.

Is it correct to say: Cancelling sin θ from a proof without checking sin θ = 0 case?

In identities we assume denominators are non-zero. Do state 'assuming sin θ ≠ 0' when clarity is needed.

Is it correct to say: Forgetting complementary-angle relationships in the exam?

sin(90° − θ) = cos θ, cot(90° − θ) = tan θ etc. Write the rule on rough sheet at the start of the paper.

Is it correct to say: Using degrees and radians together in one problem?

Class 10 uses degrees only. Do not mix degrees with π/radians until Class 11.

Introduction to Trigonometry

Q: Is it correct to say: Cancelling sin θ from a proof without checking sin θ = 0 case?

In identities we assume denominators are non-zero. Do state 'assuming sin θ ≠ 0' when clarity is needed.

Q: Is it correct to say: Forgetting complementary-angle relationships in the exam?

sin(90° − θ) = cos θ, cot(90° − θ) = tan θ etc. Write the rule on rough sheet at the start of the paper.

Q: Is it correct to say: Using degrees and radians together in one problem?

Class 10 uses degrees only. Do not mix degrees with π/radians until Class 11.

Trigonometric ratios of acute angles, values at 0°/30°/45°/60°/90°, complementary angles and identities — NCERT Class 10 Maths Ch 8

📐 10 formulas✏️ 3 examples🎯 6 practice⚖️ 6 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

💡

Board Exam Tips

→Memorise the sin/cos/tan table for 0°, 30°, 45°, 60°, 90° — half the marks come from direct substitution.
→Fundamental identities sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ. Practise the derivations.
→Prove-based questions: convert everything to sin and cos in the first step.
→Complementary angles: sin(90°−θ) = cos θ, tan(90°−θ) = cot θ, sec(90°−θ) = cosec θ.
→For sec/cosec/cot always write in terms of sin/cos to avoid confusion.

📐 Formulas(10)

Sine of an Angle★ Board fav

sin θ = \frac{opposite}{hypotenuse}

Cosine of an Angle

cos θ = \frac{adjacent}{hypotenuse}

Tangent of an Angle★ Board fav

tan θ = \frac{opposite}{adjacent} = \frac{s i n θ}{c o s θ}

Reciprocal Ratios

cosec θ = \frac{1}{s i n θ}, sec θ = \frac{1}{c o s θ}, cot θ = \frac{1}{t a n θ} = \frac{c o s θ}{s i n θ}

Pythagorean Identity (Sine-Cosine)★ Board fav

sin^{2} θ + cos^{2} θ = 1

Identity: 1 + tan²

1 + tan^{2} θ = sec^{2} θ

Identity: 1 + cot²

1 + cot^{2} θ = cosec^{2} θ

Complementary Angle Identities

sin (9 0^{\circ} - θ) = cos θ, tan (9 0^{\circ} - θ) = cot θ, sec (9 0^{\circ} - θ) = cosec θ

Standard Angle Values (Sine)★ Board fav

sin 0^{\circ} = 0, sin 3 0^{\circ} = \frac{1}{2}, sin 4 5^{\circ} = \frac{1}{2}, sin 6 0^{\circ} = \frac{3}{2}, sin 9 0^{\circ} = 1

Standard Angle Values (Tangent)

tan 0^{\circ} = 0, tan 3 0^{\circ} = \frac{1}{3}, tan 4 5^{\circ} = 1, tan 6 0^{\circ} = 3, tan 9 0^{\circ} = undefined

✏️ Solved Examples

1Solved Exampleeasy2 steps

Evaluate sin 30° + cos 60°.

Use standard values

sin 3 0^{\circ} = \frac{1}{2}, cos 6 0^{\circ} = \frac{1}{2}

2Solved Exampleboard3 steps

Prove that (1 − cos²θ) · cosec²θ = 1.

Use sin²θ + cos²θ = 1

1 - cos^{2} θ = sin^{2} θ

3Solved ExampleHOTS3 steps

Prove: (sec θ − tan θ)(sec θ + tan θ) = 1.

Expand the LHS using (a − b)(a + b) = a² − b²

LHS = sec^{2} θ - tan^{2} θ

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Confusing 'opposite' and 'adjacent' sides of the angle
✓Opposite = side facing the angle. Adjacent = side next to the angle that is NOT the hypotenuse. Hypotenuse is always across from the right angle.
2
Writing tan θ = cos θ / sin θ
✓tan θ = SIN over COS. cot θ = cos over sin. Learn the mnemonic 'SOH-CAH-TOA'.
3
Applying identities beyond 0°–90° at Class 10 level
✓Class 10 syllabus restricts θ to acute angles (0° < θ < 90°). Broader domain comes in Class 11.
4
Cancelling sin θ from a proof without checking sin θ = 0 case
✓In identities we assume denominators are non-zero. Do state 'assuming sin θ ≠ 0' when clarity is needed.
5
Forgetting complementary-angle relationships in the exam
✓sin(90° − θ) = cos θ, cot(90° − θ) = tan θ etc. Write the rule on rough sheet at the start of the paper.
6
Using degrees and radians together in one problem
✓Class 10 uses degrees only. Do not mix degrees with π/radians until Class 11.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Evaluate: 2 sin²30° + 3 tan²45°.
Q2
If tan θ = 4/3, find sin θ and cos θ.
Q3
Prove: sin θ · cot θ = cos θ.
Q4
Evaluate: cos 48° − sin 42°.
Q5
If sec θ = 13/5, find tan θ.
Q6
Prove: (1 + sin θ)/(1 − sin θ) = (sec θ + tan θ)².

📝 Notes

Introduction to Trigonometry

Trigonometry relates the angles and sides of right triangles through six ratios.

The six ratios — SOH-CAH-TOA (extended)

For a right triangle with angle θ, opposite O, adjacent A and hypotenuse H:

sin θ = O/H
cos θ = A/H
tan θ = O/A = sin/cos
cosec θ = H/O = 1/sin
sec θ = H/A = 1/cos
cot θ = A/O = cos/sin = 1/tan

Standard-angle values 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 | ∞ |

Trick: sin values 0, 1, 2, 3, 4 → divide by 4 → take √ → 0, 1/2, √2/2, √3/2, 1.