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CBSE Class 11 · Mathematics

Trigonometric Functions

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Trigonometric Functions” 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 11
  • Medium level
  • 8 concepts

1Angles

An angle at this class can be a turn measured in degrees or radians: π rad = 180°. A half-turn is π; a right angle is π/2. Angle is a turn, not a triangle-corner only — the corner is one picture of the turn.

Writing 180° as π° is a unit-mix.

Figure. A directed angle is the turn from the initial ray to the terminal ray. The same quarter-turn is 90^\circ or \pi/2 radians — two names for one picture, not two different corners.

How it works

  1. Name the turnHalf, right, full…
  2. Write it in degrees or radiansThe measure.
  3. Keep π rad = 180° as the bridgeThe angle.

2Trigonometric Functions

Trigonometric functions extend the Class 10 ratios to any real angle via the unit circle (as taught): sin, cos, tan as coordinates and their ratio. sin(π/6) is still 1/2. A function here is a rule from an angle to a number, not a triangle you must redraw every time.

Calling sin 150° “illegal” because the triangle was acute-only is a Class 10 leftover.

Figure. On the unit circle the point P at angle \theta has coordinates (\cos\theta,\sin\theta). The dashed drops are those coordinates on the axes — sine and cosine are lengths you can point at, not decorations on the triangle.

How it works

  1. Place the angle on the unit circle as taughtThe turn.
  2. Read sin, cos from the point, tan as the ratioThe functions.
  3. Reuse the stored specials with signs from the quadrantThe extend.

sin(π/6) and sin(5π/6)

Write sin(π/6) and sin(5π/6) as exact values.

  • π/61/2
  • 5π/61/2 — same sine, second quadrant as taught
  • Not−1/2 unless the cosine-sign story asked that

Pro tip. Special store plus the quadrant sign-rule.

3Trigonometric Functions of Sum and Difference of Two Angles

Sum and difference: sin(A+B) = sin A cos B + cos A sin B, and the taught twins for cos(A±B), as stored identities. They are rewrites that stay true, not a “find A” unless asked. sin(π/3 + π/6) = sin(π/2) = 1, and the expand must match.

Adding sin A + sin B as if it were sin(A+B) is the miss.

Figure. \sin(\alpha+\beta) is the sine of one combined turn, not \sin\alpha+\sin\beta. The figure keeps \alpha and the extra turn \beta on the same initial ray so the sum is a single terminal side.

How it works

  1. Write the taught expandThe identity.
  2. Substitute a pair you can storeThe check.
  3. Keep it a rewriteSum-and-difference.

4A definition is a test you can run

Angle / trig function is a test: a turn plus a named sin/cos/tan. If you cannot name the unit or the quadrant sign, you have a heading.

A pretty sine-wave poster is not the test.

Figure. The definition is a measurement test: name \theta, then form opposite/hypotenuse. Here that test reads 3/5, so \sin\theta=3/5. The words sine, cosine, tangent are names of those tests, not labels you stick on any side.

How it works

  1. Name the turn and the unitThe object.
  2. Give the value or the expandThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given is the angle (and A, B if a sum). The unknown is the value. Copy radians versus degrees before you plug π/6 into a degree-mode head.

Using 30 in place of π/6 inside a radian identity without converting is a silent given.

Figure. Write the given first: opposite =3, hypotenuse =5. Only then ask for \theta. \sin\theta=3/5 is the test that uses both givens; hunting for an adjacent you were not given is the usual stall.

How it works

  1. Copy the measure and the unitThe given.
  2. Name sin/cos/tan or the expandThe unknown.
  3. Then store or expandGiven first.

6One worked case is enough at this class

One special or one expand-check is enough. A page of the same 1/2 clone does not add a new A+B.

Ten twins of sin(π/6) are still one store.

Figure. One exact case you keep: sides 1, \sqrt{3}, 2 opposite 30^\circ, 60^\circ, 90^\circ. Then \sin 30^\circ=1/2 and \cos 30^\circ=\sqrt{3}/2 are read off the picture, not recalled as a chant.

How it works

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

7A miss: swapping the school name for the picture

A Ferris-wheel photo is a setting. The maths is a turn and a coordinate. A smile is not sin θ.

A hillside still is not A+B.

Figure. The miss is naming a side “opposite” because the word sine is in the question, not because it faces \theta. Opposite is the side you do not touch when you stand at the named angle. The adjacent is the other leg, even if it looks longer.

How it works

  1. Keep the wheel as a storyA setting.
  2. Write the angle and the functionThe maths.
  3. Then the value or the expandDo not swap.

8Check by the opposite action or the opposite test

Check a sum identity by a stored pair (π/3, π/6); check a quadrant value by the complement or a unit-circle sketch. sin(A+B) must match sin of the summed angle when both are known.

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

Figure. The opposite of \theta is the adjacent of 90^\circ-\theta. That is why \sin\theta=\cos(90^\circ-\theta): same side, two names. Checking a sine value by reading cosine of the complement is the opposite test, not a new formula to memorise.

How it works

  1. State 1 or 1/2 or the expandThe claim.
  2. Rebuild from the summed angle or the circleThe check.
  3. Mend if they disagreeHonest.
sin(A+B) equals
  1. sin A cos B + cos A sin B
  2. sin A + sin B
  3. sin A sin B

A rewrite, not a raw add.

Notes

  • The official chapter title is “Trigonometric Functions”. 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

  • π rad = 180°
  • sin(A+B)=sin A cos B + cos A sin B
  • sin(π/6)=1/2

Recap

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

Angles
An angle at this class can be a turn measured in degrees or radians: π rad = 180°.
Trigonometric Functions
Trigonometric functions extend the Class 10 ratios to any real angle via the unit circle (as taught): sin, cos, tan as coordinates and their ratio.
Trigonometric Functions of Sum and Difference of Two Angles
Sum and difference: sin(A+B) = sin A cos B + cos A sin B, and the taught twins for cos(A±B), as stored identities.
A definition is a test you can run
Angle / trig function is a test: a turn plus a named sin/cos/tan.
Name the given before the unknown
The given is the angle (and A, B if a sum).
One worked case is enough at this class
One special or one expand-check is enough.

Practise Trigonometric Functions

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