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CBSE Class 10 · Mathematics

Introduction to Trigonometry

Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.

This lesson follows the official chapter “Introduction to Trigonometry” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 10
  • Medium level
  • 8 concepts

1Trigonometric Ratios

Trigonometric ratios of an acute angle in a right triangle are opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent — sine, cosine, tangent as taught. The angle names the corner; swap the angle and opposite/adjacent swap. A ratio is two sides, not a lonely length.

Calling the hypotenuse “opposite” because it looks long is a miss.

Figure. sin θ is opp/hyp, cos θ is adj/hyp, tan θ is opp/adj. The names sit on this triangle, not on a memorised list.

How it works

  1. Mark the angle, then opposite, adjacent, hypotenuseThe names.
  2. Write sin, cos, tan as those ratiosThe trig ratios.
  3. Keep the angle in the name — sin B is not sin AThe corner matters.

2Trigonometric Ratios of Some Specific Angles

Special angles 30°, 45°, 60° have exact stored values as taught: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, and the matching cos/tan. Store them; do not freehand a decimal and call it exact. 90° and 0° as taught if the lesson included them.

sin 30° = 30/90 is not this store.

Figure. A 30-60-90 triangle stores the exact sides 1, √3, 2. sin 30° = 1/2 and tan 30° = 1/√3 come from this picture, not from a chant.

How it works

  1. Name the angle 30, 45, or 60The special.
  2. Write the stored exact valueThe ratio.
  3. Refuse a calculator-ish 0.5 guess for 45°Exact means the store.

sin 30° and tan 45°

Write sin 30° and tan 45° as exact values.

  • sin 30°1/2
  • tan 45°1
  • CheckNot 0.5 for 45°

Pro tip. Special angles are stored exact writes, not doodle decimals.

3Trigonometric Identities

An identity is a rewrite that stays true for all allowed angles: sin²θ + cos²θ = 1 is the school spine. Other taught rewrites (1 + tan²θ = sec²θ if named) must hold after a substitute. An identity is not an equation you solve for one θ.

Checking only θ = 0° can hide a fake.

Figure. sin²θ + cos²θ = 1 is Pythagoras after both sides are divided by hyp². At 30°, 1/4 + 3/4 = 1.

How it works

  1. Write the claimed identityThe rewrite.
  2. Substitute one unused allowed angle as a checkThe test.
  3. Keep it a rewrite, not a “find θ”Identity.

4A definition is a test you can run

A trig ratio is a test: a named angle in a right triangle plus two named sides. If you cannot name opposite versus hypotenuse, you have a fraction-mood.

A pretty sin-badge is not the test.

Figure. A definition is a measurement on this triangle. If you cannot point at opp and hyp, you do not yet have sin θ.

How it works

  1. Mark the angle and the three sidesThe object.
  2. Write the two-side ratioThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given is the angle and the sides (or a special angle). The unknown is a ratio or a side. Copy “opposite to A” before you divide.

Using 10 as hypotenuse because it is the largest written number, when 10 was a leg, is a silent given.

Figure. Write the given first: the 30° angle and the hypotenuse 10. The unknown opposite side is 10 × sin 30° = 5. Do not hunt a formula before the given are named.

How it works

  1. Copy the marksThe given.
  2. Name the ratio or the missing sideThe unknown.
  3. Then write sin/cos/tanGiven first.

6One worked case is enough at this class

One ratio or one identity-check is enough. A page of the same 1/2 clone does not add a new 45°.

Ten twins of sin 30° are still one store.

Figure. One worked 5-12-13 triangle is enough at this class: sin θ = 5/13, cos θ = 12/13. A second triangle that is similar does not teach a new ratio.

How it works

  1. Write one ratio or one identityThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A hillside photo is a setting. The maths is a right triangle and a named corner. A slope you “feel” is not tan 30°.

A roof still is not sin A until sides are named.

Figure. The school names opp and adj are about the named angle, not about “the upright side”. Swapping them on the picture is the usual miss.

How it works

  1. Keep the hill as a storyA setting.
  2. Draw the right triangle and name the angleThe maths.
  3. Then the ratioDo not swap.

8Check by the opposite action or the opposite test

Check a ratio by the complementary pair (sin 30° = cos 60°) if taught, or check an identity by a second angle. sin²+cos² must return 1.

A decimal that “looks close” is not the opposite test for an exact store.

Figure. Check a complementary pair on one triangle: the side opposite 30° is adjacent to 60°. So sin 30° and cos 60° are the same ratio 1/2.

How it works

  1. State 1/2 or the identityThe claim.
  2. Use the complement or another substituteThe check.
  3. Mend if they disagreeHonest.
sin 30° is
  1. 1/2
  2. √3/2
  3. 1

A stored exact value, not 30/90.

Notes

  • The official chapter title is “Introduction to Trigonometry”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • sin=opp/hyp, cos=adj/hyp, tan=opp/adj
  • sin²θ+cos²θ=1
  • sin 30°=1/2, tan 45°=1 (as taught)

Recap

Hold these pegs from the official chapter “Introduction to Trigonometry”. The wording is ExamMaster’s teaching, not a textbook recap.

Trigonometric Ratios
Trigonometric ratios of an acute angle in a right triangle are opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent — sine, cosine, tangent as taught.
Trigonometric Ratios of Some Specific Angles
Special angles 30°, 45°, 60° have exact stored values as taught: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, and the matching cos/tan.
Trigonometric Identities
An identity is a rewrite that stays true for all allowed angles: sin²θ + cos²θ = 1 is the school spine.
A definition is a test you can run
A trig ratio is a test: a named angle in a right triangle plus two named sides.
Name the given before the unknown
The given is the angle and the sides (or a special angle).
One worked case is enough at this class
One ratio or one identity-check is enough.

Practise Introduction to Trigonometry

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