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CBSE · Class 11 · Mathematics

Trigonometric Functions (Ch 3: 3.3)

5 concepts · 40 practice questions with answers · Free to read

Trigonometric functions on the unit circle

From daily life

A point moving round a circular track has two coordinates at every moment: how far east and how far north. Those two numbers are cos and sin.

Simple definition

On the unit circle (radius 1, centre at the origin), the point P at angle x has coordinates (cos x, sin x). Then tan x = sin x / cos x (cos x ≠ 0), cosec x = 1/sin x, sec x = 1/cos x, cot x = cos x / sin x.

Formula: cos x = x-coordinate sin x = y-coordinate

xy60°

In the figure: At 60°, the point is (1/2, √3/2): cos 60° = 1/2 and sin 60° = √3/2.

Rule 1: Read from coordinates — If P(a, b) is on the unit circle at angle x, then cos x = a and sin x = b.

  • Easy: P(0, 1): x = 90°, sin = 1, cos = 0.
  • Medium: P(−1, 0): x = 180°, sin = 0, cos = −1.
  • Exam-type: P(3/5, 4/5): cos x = 3/5, sin x = 4/5, tan x = 4/3.

Rule 2: Reciprocal functions — Flip the value.

  • Easy: If sin x = 1/2, cosec x = 2.
  • Medium: If cos x = 3/5, sec x = 5/3.
  • Exam-type: If tan x = 2/3, cot x = 3/2.
Remember

Since the radius is 1: sin²x + cos²x = 1.

Common mistake

Mixing up which coordinate is sin and which is cos. Cos = x (horizontal), Sin = y (vertical).

Memory clue

Alphabetical order: C comes before S, just as x comes before y.

Practice questions

Easy
  1. 1P(1, 0) is at angle 0°. Find sin 0° and cos 0°.
  2. 2Find tan 45° from P(1/√2, 1/√2).
  3. 3If sin x = 3/5 and cos x = 4/5, find tan x.
Medium
  1. 4If cos x = 5/13, find sec x.
  2. 5Check that P(5/13, 12/13) is on the unit circle.
  3. 6Find cot x if sin x = 12/13 and cos x = 5/13.
Challenging
  1. 7If sin x = 0.6 on the unit circle in quadrant I, find cos x.
  2. 8If cos x = −3/5 in quadrant II, find sin x.
Show answers (try them yourself first)
  1. 1sin 0° = 0, cos 0° = 1
  2. 21
  3. 33/4
  4. 413/5
  5. 5(5/13)² + (12/13)² = 25/169 + 144/169 = 169/169 = 1 ✓
  6. 65/12
  7. 7√(1 − 0.36) = 0.8
  8. 84/5

Signs in the four quadrants

From daily life

North-east, north-west, south-west, south-east: the sign of east–west and north–south positions changes from quarter to quarter.

Simple definition

The signs of sin x and cos x depend on the sign of the y and x coordinates. I: all positive; II: only sin and cosec positive; III: only tan and cot positive; IV: only cos and sec positive.

I : all +II : only sin, cosec +III : only tan, cot +IV : only cos, sec +

In the figure: Read the quadrant, then which functions stay positive.

Rule 1: Find the sign — Find the quadrant first.

  • Easy: sin 120°: quadrant II, positive.
  • Medium: cos 200°: quadrant III, negative.
  • Exam-type: tan 300°: quadrant IV, negative.

Rule 2: Reduce to an acute angle — Use the nearest x-axis: 180° − x, x − 180° or 360° − x.

  • Easy: sin 150° = sin 30° = 1/2.
  • Medium: cos 120° = −cos 60° = −1/2.
  • Exam-type: tan 225° = tan 45° = 1.
Remember

Quadrant I: All. II: Sin. III: Tan. IV: Cos.

Common mistake

Forgetting the sign when the angle is reduced to an acute one.

Memory clue

All Students Take Chocolates (I: All, II: Sin, III: Tan, IV: Cos).

Practice questions

Easy
  1. 1Sign of sin 250°?
  2. 2Sign of cos 330°?
  3. 3Sign of tan 100°?
Medium
  1. 4Find sin 210°.
  2. 5Find cos 300°.
  3. 6Find tan 135°.
Challenging
  1. 7If sin x > 0 and cos x < 0, in which quadrant is x?
  2. 8If tan x > 0 and sin x < 0, in which quadrant is x?
Show answers (try them yourself first)
  1. 1Quadrant III, negative
  2. 2Quadrant IV, positive
  3. 3Quadrant II, negative
  4. 4−sin 30° = −1/2
  5. 5cos 60° = 1/2
  6. 6−tan 45° = −1
  7. 7Quadrant II
  8. 8Quadrant III

Values for standard angles

From daily life

A carpenter keeps a set-square with 30°-60° and 45° corners. The numbers they give are used everywhere.

Simple definition

Learn this table; the other values are obtained from it with signs.

Rule 1: Use the table — Read the value and fix the sign by the quadrant.

  • Easy: sin 30° = 1/2, cos 60° = 1/2.
  • Medium: sin 45° = cos 45° = 1/√2.
  • Exam-type: sin 60° = √3/2, tan 60° = √3.

Rule 2: Combine values — Replace each part by its table value.

  • Easy: sin 30° + cos 60° = 1/2 + 1/2 = 1.
  • Medium: sin²45° + cos²45° = 1/2 + 1/2 = 1.
  • Exam-type: tan 60° × tan 30° = √3 × 1/√3 = 1.
Remember

sin and cos are swapped between 30° and 60°: sin 30° = cos 60°.

Common mistake

Writing tan 90° = 1. It is not defined, because cos 90° = 0.

Memory clue

sin: 0, 1/2, 1/√2, √3/2, 1 → √0/2, √1/2, √2/2, √3/2, √4/2.

Quick table

angle0°30°45°60°90°
sin01/21/√2√3/21
cos1√3/21/√21/20
tan01/√31√3not defined

Practice questions

Easy
  1. 1Find sin 60° × cos 30°.
  2. 2Find tan 45° + sin 90°.
  3. 3Find cos 0° + sin 0°.
Medium
  1. 4Find 2 sin 30° + 3 cos 60°.
  2. 5Find sin 45° × cos 45°.
  3. 6Find (sin 30° + cos 60°)².
Challenging
  1. 7Find the value of 4 tan²45° − sin²60°.
  2. 8Find cos 60° cos 30° − sin 60° sin 30°.
Show answers (try them yourself first)
  1. 1(√3/2)(√3/2) = 3/4
  2. 22
  3. 31
  4. 41 + 1.5 = 2.5
  5. 51/2
  6. 61
  7. 74 − 3/4 = 3.25
  8. 8(1/2)(√3/2) − (√3/2)(1/2) = 0

Domain and range of the six functions

From daily life

A thermometer reading rises and falls but never goes beyond its maximum and minimum. sin and cos do the same between −1 and 1.

Simple definition

sin x and cos x are defined for every real x and lie between −1 and 1. tan x and sec x are not defined where cos x = 0 (x = π/2, 3π/2, …). cot x and cosec x are not defined where sin x = 0 (x = 0, π, 2π, …). tan x and cot x can take any real value; sec x and cosec x never lie strictly between −1 and 1.

Rule 1: Range — sin and cos: [−1, 1]. tan and cot: R. sec and cosec: (−∞, −1] ∪ [1, ∞).

  • Easy: 2 sin x has range [−2, 2].
  • Medium: 3 cos x + 1 has range [−2, 4].
  • Exam-type: sin x = 2 is impossible.

Rule 2: Period — sin, cos, sec, cosec repeat every 2π; tan and cot repeat every π.

  • Easy: sin(x + 2π) = sin x.
  • Medium: tan(x + π) = tan x.
  • Exam-type: cos 390° = cos 30° = √3/2.
Remember

sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x.

Common mistake

Saying sin x can be 3/2 for some x.

Memory clue

Sin and Cos live inside the strip [−1, 1].

Practice questions

Easy
  1. 1Is sin x = 1.5 possible?
  2. 2Find the range of 5 sin x.
  3. 3Find the range of cos x − 2.
Medium
  1. 4At which x in [0, 2π] is tan x not defined?
  2. 5Find the range of 4 − 3 sin x.
  3. 6Find sin 765°.
Challenging
  1. 7If sin x = 1/3 in quadrant I, find cos x and tan x.
  2. 8If tan x = 3/4 in quadrant III, find sin x and cos x.
Show answers (try them yourself first)
  1. 1No — sin x never exceeds 1.
  2. 2[−5, 5]
  3. 3[−3, −1]
  4. 4π/2 and 3π/2
  5. 5[1, 7]
  6. 6765 = 2 × 360 + 45, so sin 45° = 1/√2
  7. 7cos x = √(1 − 1/9) = 2√2/3, tan x = 1/(2√2)
  8. 8sin x = −3/5, cos x = −4/5

Basic identities

From daily life

Like an exchange rate that always holds, an identity is a statement true for every allowed angle.

Simple definition

Because the point (cos x, sin x) lies on the unit circle: sin²x + cos²x = 1. Dividing by cos²x gives 1 + tan²x = sec²x; dividing by sin²x gives 1 + cot²x = cosec²x.

Formula: sin²x + cos²x = 1

Rule 1: Find one ratio from another — Use the identity and fix the sign by the quadrant.

  • Easy: sin x = 3/5 (quadrant I): cos²x = 16/25, cos x = 4/5.
  • Medium: tan x = 1 (quadrant I): sec x = √2.
  • Exam-type: cos x = −1/2 (quadrant II): sin x = √3/2.

Rule 2: Prove an identity — Change everything to sin and cos, then simplify.

  • Easy: (1 − sin x)(1 + sin x) = cos²x.
  • Medium: sec x − cos x = sin x tan x.
  • Exam-type: tan x + cot x = sec x cosec x.
Remember

In proofs, work on the more complicated side only.

Common mistake

Cancelling terms across a sum, such as (sin x + cos x)/cos x = sin x + 1 (wrong).

Memory clue

Pythagoras on the unit circle.

Practice questions

Easy
  1. 1Simplify 1 − sin²x.
  2. 2Simplify sec²x − tan²x.
  3. 3Simplify (sin x + cos x)² − 2 sin x cos x.
Medium
  1. 4If cos x = 12/13 in quadrant IV, find sin x.
  2. 5If tan x = −5/12 in quadrant II, find sin x and cos x.
  3. 6Prove: sin x cosec x = 1.
Challenging
  1. 7Simplify (1 + tan²x) cos²x.
  2. 8Prove: 1/(1 + sin x) + 1/(1 − sin x) = 2 sec²x.
Show answers (try them yourself first)
  1. 1cos²x
  2. 21
  3. 31
  4. 4−5/13
  5. 5sin x = 5/13, cos x = −12/13
  6. 6cosec x = 1/sin x, so the product is 1.
  7. 7sec²x · cos²x = 1
  8. 8(1 − sin x + 1 + sin x)/(1 − sin²x) = 2/cos²x