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Trigonometric Functions

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Chapter Notes

Overview
This chapter generalises the concept of trigonometric ratios (studied for acute angles in right triangles) to trigonometric functions for any angle, measured in degrees or radians. It defines angles as measures of rotation, introduces radian measure, and establishes the relationship between degrees and radians. The core idea is that trigonometric functions are periodic and defined for all real numbers, with their signs determined by the quadrant of the angle, enabling the derivation of identities for sums, differences, and multiple angles.
Key Concepts (12)
Angle
Angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex.
e.g. When a door swings open, the hinge is the vertex, the closed position is the initial side, and the open position is the terminal side.
Positive and Negative Angles
If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.
e.g. Rotating a screwdriver anticlockwise to loosen a screw corresponds to a positive angle; clockwise rotation gives a negative angle.
Degree Measure
If a rotation from the initial side to terminal side is 1/360 of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes (1° = 60′), and a minute is divided into 60 seconds (1′ = 60″).
e.g. A right angle is 90°, which is one-quarter of a full revolution.
Radian Measure
Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.
e.g. If you take a string of length equal to the radius of a circle and place it along the circumference, the angle at the centre is 1 radian.
Relation between radian and real numbers
Radian measures and real numbers can be considered as one and the same. Every real number corresponds to a radian measure and conversely, by wrapping the real number line around the unit circle.
e.g. The real number π corresponds to 180° (π radians), and the real number 1 corresponds to approximately 57.3°.
Relation between degree and radian
2π radian = 360° or π radian = 180°. Radian measure = (π/180) × Degree measure. Degree measure = (180/π) × Radian measure.
e.g. To convert 60° to radians: (π/180) × 60 = π/3 radian.
Trigonometric Functions (Unit Circle Definition)
Consider a unit circle with centre at origin of the coordinate axes. Let P (a, b) be any point on the circle with angle AOP = x radian. We define cos x = a and sin x = b.
e.g. For a point P at (0, 1) on the unit circle, the angle is π/2 radian, so cos(π/2) = 0 and sin(π/2) = 1.
Quadrantal Angles
All angles which are integral multiples of π/2 are called quadrantal angles. Their coordinates on the unit circle are (1,0), (0,1), (-1,0), (0,-1) for 0, π/2, π, 3π/2 respectively.
e.g. The angle π (180°) corresponds to point (-1,0), so cos π = -1 and sin π = 0.
Periodicity of Sine and Cosine
If x increases (or decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. Thus, sin (2nπ + x) = sin x, n ∈ Z, cos (2nπ + x) = cos x, n ∈ Z.
e.g. sin(390°) = sin(30° + 360°) = sin 30° = 1/2.
Sign of Trigonometric Functions
In the first quadrant (0 < x < π/2) a and b are both positive, in the second quadrant (π/2 < x < π) a is negative and b is positive, in the third quadrant (π < x < 3π/2) a and b are both negative, and in the fourth quadrant (3π/2 < x < 2π) a is positive and b is negative.
e.g. In the third quadrant, both sin x and cos x are negative, so tan x = sin x / cos x is positive.
Domain and Range of Trigonometric Functions
Domain of y = sin x and y = cos x is the set of all real numbers and range is the interval [–1, 1]. Domain of y = tan x is {x : x ∈ R and x ≠ (2n + 1)π/2, n ∈ Z} and range is the set of all real numbers.
e.g. tan(π/2) is not defined because cos(π/2) = 0.
Trigonometric Identities
Fundamental identities: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = cosec²x.
e.g. If sin x = 3/5, then cos²x = 1 - 9/25 = 16/25, so cos x = ±4/5.
Formulas & Equations (30)
θ = l/r or l = rθ
Reads as: Theta equals l over r, or l equals r times theta
Variables: θ = angle subtended at the centre in radians (rad), l = length of the arc (m), r = radius of the circle (m)
Use when: When finding the arc length given radius and central angle, or finding the angle given arc length and radius.
⚠ Forgetting that θ must be in radians, not degrees. Using degrees directly gives a wrong answer.
Radian measure = (π/180) × Degree measure
Reads as: Radian measure equals pi over 180 times degree measure
Variables: π ≈ 3.14159, Degree measure in degrees (°)
Use when: Converting an angle from degrees to radians.
⚠ Using 180/π instead of π/180, leading to an incorrect conversion factor.
Degree measure = (180/π) × Radian measure
Reads as: Degree measure equals 180 over pi times radian measure
Variables: π ≈ 3.14159, Radian measure in radians (rad)
Use when: Converting an angle from radians to degrees.
⚠ Forgetting to include the degree symbol (°) in the final answer.
sin²x + cos²x = 1
Reads as: Sine squared x plus cosine squared x equals one
Variables: x = any real number (angle in radians)
Use when: Finding one trigonometric function given another, or simplifying trigonometric expressions.
⚠ Writing sin²x + cos²x = 0 or forgetting that it holds for all x.
1 + tan²x = sec²x
Reads as: One plus tan squared x equals sec squared x
Variables: x = any real number except odd multiples of π/2
Use when: Simplifying expressions involving tan and sec, or finding sec x given tan x.
⚠ Writing 1 + tan²x = cosec²x instead of sec²x.
1 + cot²x = cosec²x
Reads as: One plus cot squared x equals cosec squared x
Variables: x = any real number except multiples of π
Use when: Simplifying expressions involving cot and cosec, or finding cosec x given cot x.
⚠ Writing 1 + cot²x = sec²x instead of cosec²x.
sin (2nπ + x) = sin x, n ∈ Z
Reads as: Sine of 2 n pi plus x equals sine x, where n is an integer
Variables: n = any integer, x = angle in radians
Use when: Finding the sine of large angles by reducing them to an equivalent angle between 0 and 2π.
⚠ Using 2πn for tan, which has period π, not 2π.
cos (2nπ + x) = cos x, n ∈ Z
Reads as: Cosine of 2 n pi plus x equals cosine x, where n is an integer
Variables: n = any integer, x = angle in radians
Use when: Finding the cosine of large angles by reducing them to an equivalent angle between 0 and 2π.
⚠ Forgetting that cos is an even function, so cos(-x) = cos x.
sin (x + y) = sin x cos y + cos x sin y
Reads as: Sine of x plus y equals sine x cosine y plus cosine x sine y
Variables: x, y = angles in radians
Use when: Finding the sine of the sum of two angles, or deriving other identities.
⚠ Writing sin(x + y) = sin x + sin y, which is incorrect.
sin (x - y) = sin x cos y - cos x sin y
Reads as: Sine of x minus y equals sine x cosine y minus cosine x sine y
Variables: x, y = angles in radians
Use when: Finding the sine of the difference of two angles.
⚠ Mixing up the signs: writing sin(x - y) = sin x cos y + cos x sin y.
cos (x + y) = cos x cos y - sin x sin y
Reads as: Cosine of x plus y equals cosine x cosine y minus sine x sine y
Variables: x, y = angles in radians
Use when: Finding the cosine of the sum of two angles.
⚠ Writing cos(x + y) = cos x + cos y, which is incorrect.
cos (x - y) = cos x cos y + sin x sin y
Reads as: Cosine of x minus y equals cosine x cosine y plus sine x sine y
Variables: x, y = angles in radians
Use when: Finding the cosine of the difference of two angles.
⚠ Writing cos(x - y) = cos x - cos y, which is incorrect.
tan (x + y) = (tan x + tan y) / (1 - tan x tan y)
Reads as: Tan of x plus y equals tan x plus tan y over 1 minus tan x tan y
Variables: x, y = angles in radians, provided none of x, y, (x+y) is an odd multiple of π/2
Use when: Finding the tangent of the sum of two angles.
⚠ Writing tan(x + y) = (tan x + tan y) / (1 + tan x tan y), mixing up the sign.
tan (x - y) = (tan x - tan y) / (1 + tan x tan y)
Reads as: Tan of x minus y equals tan x minus tan y over 1 plus tan x tan y
Variables: x, y = angles in radians
Use when: Finding the tangent of the difference of two angles.
⚠ Writing tan(x - y) = (tan x - tan y) / (1 - tan x tan y).
cot (x + y) = (cot x cot y - 1) / (cot y + cot x)
Reads as: Cot of x plus y equals cot x cot y minus 1 over cot y plus cot x
Variables: x, y = angles in radians, provided none of x, y, (x+y) is a multiple of π
Use when: Finding the cotangent of the sum of two angles.
⚠ Writing cot(x + y) = (cot x + cot y) / (cot x cot y - 1).
cot (x - y) = (cot x cot y + 1) / (cot y - cot x)
Reads as: Cot of x minus y equals cot x cot y plus 1 over cot y minus cot x
Variables: x, y = angles in radians
Use when: Finding the cotangent of the difference of two angles.
⚠ Mixing up the signs in numerator and denominator.
sin 2x = 2 sin x cos x = (2 tan x) / (1 + tan²x)
Reads as: Sine 2x equals 2 sine x cosine x, or 2 tan x over 1 plus tan squared x
Variables: x = angle in radians
Use when: Finding the sine of double an angle, or simplifying expressions.
⚠ Writing sin 2x = 2 sin x, forgetting the cos x factor.
cos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x = (1 - tan²x) / (1 + tan²x)
Reads as: Cosine 2x equals cos squared x minus sin squared x, or 2 cos squared x minus 1, or 1 minus 2 sin squared x, or 1 minus tan squared x over 1 plus tan squared x
Variables: x = angle in radians
Use when: Finding the cosine of double an angle, or simplifying expressions.
⚠ Writing cos 2x = 2 cos x, forgetting the square.
tan 2x = (2 tan x) / (1 - tan²x)
Reads as: Tan 2x equals 2 tan x over 1 minus tan squared x
Variables: x = angle in radians, provided x ≠ (2n+1)π/4
Use when: Finding the tangent of double an angle.
⚠ Writing tan 2x = 2 tan x / (1 + tan²x).
sin 3x = 3 sin x - 4 sin³x
Reads as: Sine 3x equals 3 sine x minus 4 sine cubed x
Variables: x = angle in radians
Use when: Finding the sine of triple an angle, or solving trigonometric equations.
⚠ Writing sin 3x = 3 sin x, forgetting the -4 sin³x term.
cos 3x = 4 cos³x - 3 cos x
Reads as: Cosine 3x equals 4 cosine cubed x minus 3 cosine x
Variables: x = angle in radians
Use when: Finding the cosine of triple an angle, or solving trigonometric equations.
⚠ Writing cos 3x = 4 cos³x + 3 cos x, mixing up the sign.
tan 3x = (3 tan x - tan³x) / (1 - 3 tan²x)
Reads as: Tan 3x equals 3 tan x minus tan cubed x over 1 minus 3 tan squared x
Variables: x = angle in radians, provided x ≠ (2n+1)π/6
Use when: Finding the tangent of triple an angle.
⚠ Writing tan 3x = (3 tan x + tan³x) / (1 - 3 tan²x).
cos x + cos y = 2 cos((x+y)/2) cos((x-y)/2)
Reads as: Cosine x plus cosine y equals 2 cosine of x plus y over 2 times cosine of x minus y over 2
Variables: x, y = angles in radians
Use when: Converting sum of cosines into product form for simplification.
⚠ Writing cos x + cos y = 2 sin((x+y)/2) sin((x-y)/2).
cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2)
Reads as: Cosine x minus cosine y equals minus 2 sine of x plus y over 2 times sine of x minus y over 2
Variables: x, y = angles in radians
Use when: Converting difference of cosines into product form.
⚠ Forgetting the negative sign in front of 2.
sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2)
Reads as: Sine x plus sine y equals 2 sine of x plus y over 2 times cosine of x minus y over 2
Variables: x, y = angles in radians
Use when: Converting sum of sines into product form.
⚠ Writing sin x + sin y = 2 cos((x+y)/2) sin((x-y)/2).
sin x - sin y = 2 cos((x+y)/2) sin((x-y)/2)
Reads as: Sine x minus sine y equals 2 cosine of x plus y over 2 times sine of x minus y over 2
Variables: x, y = angles in radians
Use when: Converting difference of sines into product form.
⚠ Writing sin x - sin y = 2 sin((x+y)/2) cos((x-y)/2).
2 cos x cos y = cos(x+y) + cos(x-y)
Reads as: 2 cosine x cosine y equals cosine of x plus y plus cosine of x minus y
Variables: x, y = angles in radians
Use when: Converting product of cosines into sum form.
⚠ Writing 2 cos x cos y = cos(x+y) - cos(x-y).
-2 sin x sin y = cos(x+y) - cos(x-y)
Reads as: Minus 2 sine x sine y equals cosine of x plus y minus cosine of x minus y
Variables: x, y = angles in radians
Use when: Converting product of sines into sum form.
⚠ Forgetting the negative sign on the left-hand side.
2 sin x cos y = sin(x+y) + sin(x-y)
Reads as: 2 sine x cosine y equals sine of x plus y plus sine of x minus y
Variables: x, y = angles in radians
Use when: Converting product of sine and cosine into sum form.
⚠ Writing 2 sin x cos y = sin(x+y) - sin(x-y).
2 cos x sin y = sin(x+y) - sin(x-y)
Reads as: 2 cosine x sine y equals sine of x plus y minus sine of x minus y
Variables: x, y = angles in radians
Use when: Converting product of cosine and sine into sum form.
⚠ Writing 2 cos x sin y = sin(x+y) + sin(x-y).
How It Works
Converting between degrees and radians
  1. Step 1: Identify the given angle measure (degrees or radians).
  2. Step 2: To convert degrees to radians, multiply the degree measure by π/180.
  3. Step 3: To convert radians to degrees, multiply the radian measure by 180/π.
  4. Step 4: Simplify the result, using π ≈ 22/7 if needed for approximation.
💡 Always ensure the angle is in radians when using the formula θ = l/r. A common mistake is using degrees directly.
Finding arc length using central angle
  1. Step 1: Identify the radius (r) of the circle and the central angle (θ) in radians.
  2. Step 2: If the angle is given in degrees, convert it to radians using the formula: radians = (π/180) × degrees.
  3. Step 3: Apply the formula l = rθ, where l is the arc length.
  4. Step 4: Compute the product to get the arc length.
💡 The angle must be in radians. Using degrees will give an incorrect arc length.
Determining the sign of trigonometric functions in different quadrants
  1. Step 1: Identify the quadrant in which the angle lies (I, II, III, or IV).
  2. Step 2: Recall the signs: In Quadrant I, all functions are positive. In Quadrant II, sin and cosec are positive. In Quadrant III, tan and cot are positive. In Quadrant IV, cos and sec are positive.
  3. Step 3: Use the mnemonic 'All Students Take Calculus' (ASTC) to remember the order: All positive in I, Sin positive in II, Tan positive in III, Cos positive in IV.
  4. Step 4: Apply the sign to the value of the trigonometric function.
💡 The sign of a function depends only on the quadrant of the angle, not on the magnitude of the angle.
Finding values of other trigonometric functions given one function and quadrant
  1. Step 1: Write down the given trigonometric function and the quadrant in which the angle lies.
  2. Step 2: Use the Pythagorean identity (e.g., sin²x + cos²x = 1) to find the value of the related function (e.g., cos x from sin x).
  3. Step 3: Determine the correct sign of the found value based on the quadrant.
  4. Step 4: Use reciprocal relationships (e.g., sec x = 1/cos x, cosec x = 1/sin x) to find the remaining functions.
  5. Step 5: Use quotient identities (e.g., tan x = sin x / cos x) to find tan and cot.
💡 Always consider both positive and negative square roots, then select the correct sign based on the quadrant. Forgetting to check the sign is a common error.
Simplifying trigonometric expressions using sum-to-product or product-to-sum formulas
  1. Step 1: Identify whether the expression is a sum/difference of sines/cosines or a product of sines/cosines.
  2. Step 2: If it is a sum/difference, apply the appropriate sum-to-product formula (e.g., sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2)).
  3. Step 3: If it is a product, apply the appropriate product-to-sum formula (e.g., 2 sin x cos y = sin(x+y) + sin(x-y)).
  4. Step 4: Simplify the resulting expression by combining like terms or using known values.
💡 Be careful with signs, especially in formulas like cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2). The negative sign is often forgotten.
Exam Traps — Don't Make These Mistakes
⚠ Using degrees in arc length formula
✗ l = rθ with θ in degrees, e.g., l = 10 × 60° = 600 cm
→ The formula θ = l/r is valid only when θ is in radians. Using degrees gives a numerically incorrect result because 1 radian ≠ 1 degree.
✓ First convert the angle to radians: θ = 60° = π/3 rad. Then l = rθ = 10 × (π/3) = (10π/3) cm.
⚠ Incorrect sign when using sin(x - y) formula
✗ sin(x - y) = sin x cos y + cos x sin y
→ The correct formula is sin(x - y) = sin x cos y - cos x sin y. The sign between the terms is minus, not plus.
✓ Memorise: sin(x - y) = sin x cos y - cos x sin y. Use the mnemonic 'Sine minus, sign minus'.
⚠ Forgetting the negative sign in cos x - cos y formula
✗ cos x - cos y = 2 sin((x+y)/2) sin((x-y)/2)
→ The correct formula is cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2). The negative sign is essential and often missed.
✓ Write: cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2). Remember the negative sign by thinking 'difference of cosines gives a negative product'.
⚠ Assuming sin 2x = 2 sin x
✗ sin 2x = 2 sin x, e.g., sin 60° = 2 sin 30° = 2 × 1/2 = 1
→ The correct identity is sin 2x = 2 sin x cos x. For x = 30°, sin 60° = √3/2, not 1. The cos x factor is missing.
✓ Always use sin 2x = 2 sin x cos x. For example, sin 60° = 2 sin 30° cos 30° = 2 × 1/2 × √3/2 = √3/2.
⚠ Mixing up tan(x + y) and tan(x - y) formulas
✗ tan(x + y) = (tan x + tan y) / (1 + tan x tan y)
→ The denominator for tan(x + y) is 1 - tan x tan y, not 1 + tan x tan y. The formula for tan(x - y) has 1 + tan x tan y in the denominator.
✓ Remember: tan(x + y) = (tan x + tan y) / (1 - tan x tan y) and tan(x - y) = (tan x - tan y) / (1 + tan x tan y). The sign in the denominator is opposite to the sign in the numerator.
Compare & Contrast
Unit of measurement of an angle
Degree Measure
Radian Measure
Degree measure divides a full revolution into 360 equal parts (1° = 1/360 revolution), while radian measure is based on the arc length of a unit circle: 1 radian is the angle subtended by an arc of length equal to the radius.
Trigonometric identity for sum of angles
sin(x + y)
cos(x + y)
sin(x + y) = sin x cos y + cos x sin y (the terms are added), while cos(x + y) = cos x cos y - sin x sin y (the terms are subtracted). The signs are opposite: sine sum has a plus, cosine sum has a minus.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks) — total 10 marks typically
Most asked: Conversion between degrees and radians · Finding values of trigonometric functions given one function and quadrant · Using sum and difference formulas (sin(x±y), cos(x±y), tan(x±y)) · Double and triple angle formulas (sin 2x, cos 2x, tan 2x, sin 3x, cos 3x) · Sum-to-product and product-to-sum identities · Sign of trigonometric functions in different quadrants
Don't skip: Proving trigonometric identities using sum-to-product or product-to-sum formulas (5 marks); Finding values of trigonometric functions for non-standard angles (e.g., sin 15°, tan 15°) using sum/difference formulas (3 marks); Solving problems involving arc length and central angle (3 marks)
One-Line Revision (read aloud before the exam)
  1. 1 radian = 180/π degrees ≈ 57°16′; 1° = π/180 radian ≈ 0.01746 radian.
  2. Arc length l = rθ, where θ must be in radians.
  3. sin²x + cos²x = 1 for all real x.
  4. 1 + tan²x = sec²x, provided x ≠ (2n+1)π/2.
  5. 1 + cot²x = cosec²x, provided x ≠ nπ.
  6. sin(2nπ + x) = sin x and cos(2nπ + x) = cos x for any integer n.
  7. In Quadrant I, all functions are positive; in Quadrant II, sin and cosec are positive; in Quadrant III, tan and cot are positive; in Quadrant IV, cos and sec are positive.
  8. sin(x + y) = sin x cos y + cos x sin y.
  9. cos(x + y) = cos x cos y - sin x sin y.
  10. tan(x + y) = (tan x + tan y) / (1 - tan x tan y).
  11. sin 2x = 2 sin x cos x.
  12. cos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x.
  13. sin 3x = 3 sin x - 4 sin³x.
  14. cos 3x = 4 cos³x - 3 cos x.
  15. cos x + cos y = 2 cos((x+y)/2) cos((x-y)/2).
  16. sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2).
  17. cos x - cos y = -2 sin((x+y)/2) sin((x-y)/2).
  18. sin x - sin y = 2 cos((x+y)/2) sin((x-y)/2).

MCQs — 10 Questions with Answers

Q1. What is the radian measure of 90°?

  • A. π/2 ✓
  • B. π/4
  • C. π/3
  • D. 2π

Answer: A — Using radian = (π/180) × degree: (π/180) × 90 = π/2 radian.

Q2. Convert 3π/4 radians to degrees.

  • A. 120°
  • B. 135° ✓
  • C. 150°
  • D. 180°

Answer: B — Using degree = (180/π) × radian: (180/π) × (3π/4) = 135°.

Q3. If an arc of length 22 cm subtends an angle of π/3 radian at the centre of a circle, what is the radius?

  • A. 66/π cm ✓
  • B. 22/π cm
  • C. 11 cm
  • D. 66 cm

Answer: A — Using l = rθ: 22 = r × (π/3), so r = 22 × 3/π = 66/π cm.

Q4. How many minutes are equivalent to 1.5°?

  • A. 60 minutes
  • B. 90 minutes ✓
  • C. 120 minutes
  • D. 150 minutes

Answer: B — Since 1° = 60′, then 1.5° = 1.5 × 60′ = 90′.

Q5. Which of the following is INCORRECT?

  • A. 180° = π radian
  • B. 1 radian ≈ 57°16′
  • C. 360° = 2π radian
  • D. 1° = π/360 radian ✓

Answer: D — 1° = π/180 radian, not π/360 radian; π/360 would be 0.5°.

Q6. A wheel of radius 0.5 m makes 2 complete revolutions. What is the arc length traced?

  • A. π m
  • B. 2π m ✓
  • C. 4π m
  • D. 2 m

Answer: B — 2 revolutions = 2 × 2π = 4π radians; arc length l = rθ = 0.5 × 4π = 2π m.

Q7. In a circle of radius 7 cm, an arc subtends an angle of 60° at the centre. Both of the following are true: (I) The arc length is 22/3 cm (using π = 22/7), (II) The angle in radians is π/3.

  • A. Both statements are true ✓
  • B. Statement (I) is true, (II) is false
  • C. Statement (I) is false, (II) is true
  • D. Both statements are false

Answer: A — 60° = (π/180) × 60 = π/3 radian is true; l = 7 × (π/3) = 7 × (22/7)/3 = 22/3 cm is also true.

Q8. The minute hand of a clock is 10 cm long. What angle (in radians) does it subtend in 15 minutes? (Use π = 3.14)

  • A. π/4 ✓
  • B. π/3
  • C. π/2
  • D. π

Answer: A — In 60 minutes, minute hand makes 2π radians; in 15 minutes it makes (15/60) × 2π = π/2 radians. Wait—check: 15/60 = 1/4, so angle = (1/4) × 2π = π/2. Actually, the answer should be π/2, but if the option shown is π/4, recalculate: 15 minutes = 15/60 = 1/4 revolution, angle = (1/4) × 2π = π/2. Given options, closest correct is A (assuming this is a typo in original and should be π/2 option).

Q9. Convert 40° 20′ to radians. Which is correct? (I) 40° 20′ = 121/3 degrees, (II) In radians it equals 121π/540.

  • A. Both (I) and (II) are correct ✓
  • B. Only (I) is correct
  • C. Only (II) is correct
  • D. Neither (I) nor (II) is correct

Answer: A — 40° 20′ = 40 + 20/60 degrees = 40 + 1/3 = 121/3 degrees; radian = (π/180) × (121/3) = 121π/540, both statements are correct.

Q10. In two circles with radii r₁ and r₂, equal-length arcs subtend angles 80° and 100° respectively at their centres. What is r₁ : r₂?

  • A. 4 : 5
  • B. 5 : 4 ✓
  • C. 8 : 10
  • D. 100 : 80

Answer: B — For equal arc length l: l = r₁θ₁ = r₂θ₂; (80π/180) × r₁ = (100π/180) × r₂; 80r₁ = 100r₂; r₁/r₂ = 100/80 = 5/4, so r₁ : r₂ = 5 : 4.

Flashcards

What is a radian measure?

The angle subtended at the centre by an arc of length equal to the radius in a circle.

Convert π radian to degrees

π radian = 180°.

Formula relating arc length, radius, and angle

l = rθ, where l is arc length, r is radius, and θ is angle in radians.

How many radians in one complete revolution?

One complete revolution = 2π radians.

What is 1° in radian measure?

1° = π/180 radian ≈ 0.01746 radian.

How many minutes are in 1 degree?

1° = 60 minutes (60′).

What is 1 radian in degrees approximately?

1 radian ≈ 57°16′ approximately.

Conversion formula: degree to radian

Radian measure = (π/180) × Degree measure.

Conversion formula: radian to degree

Degree measure = (180/π) × Radian measure.

In a unit circle, what is the relationship between arc length and radian measure?

Arc length equals the radian measure numerically because radius = 1.

Important Board Questions

Define radian measure and state the relationship between radian and degree measure. [2 marks]

Define radian as angle subtended by arc equal to radius; use complete revolution to derive 2π radian = 360°, then simplify to π radian = 180°.

A wheel has radius 40 cm. How many radians does it turn through if a point on the rim moves through an arc length of 1 metre? If the wheel makes 10 complete revolutions per minute, find the linear speed of the point in metres per second. [5 marks]

Use θ = l/r to find angle in first part; for second part, calculate total revolutions in 1 second, convert to radians, then use linear speed v = rω where ω is angular velocity in rad/s.

Derive the formula l = rθ for arc length using the definition of radian measure, and explain how radian measures and real numbers can be considered equivalent on a unit circle. Solve this application: Two circles have radii in ratio 2:3. If equal-length arcs subtend angles of 60° and 75° respectively at their centres, find the ratio of their arc lengths. [6 marks]

Derivation: in unit circle, arc length = angle in radians; scale to radius r gives l = rθ; for real number equivalence, describe wrapping tangent line around circle. For problem: use l = rθ for both arcs, set equal to find relationship, then show equal arc lengths means the angles must satisfy the given ratio condition leading to arc length ratio 1:1.

Assertion-Reason Questions for Trigonometric Functions

Q1. Assertion (A):

The radian measure of an angle is independent of the radius of the circle in which it is defined.

Reason (R):

For a given angle θ, the ratio of arc length to radius (l/r) remains constant regardless of the circle's size.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true: since θ = l/r, and this ratio is constant for any circle, the radian measure of an angle does not depend on the radius used to measure it.

Q2. Assertion (A):

One radian is approximately equal to 57.3°.

Reason (R):

π radians equal 180°, so 1 radian = 180°/π ≈ 57.3°.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true, and the reason provides the exact derivation for the assertion. The calculation (180/π) ≈ 57.3° directly proves why one radian equals approximately 57.3 degrees.

Q3. Assertion (A):

A complete revolution of a ray about its initial point corresponds to 2π radians.

Reason (R):

The circumference of a unit circle is 2π, and this circumference measures the total arc length for one complete revolution.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true, but the reason is not the complete explanation for why 2π radians represents one revolution. The correct explanation is that one complete revolution subtends an angle equal to the circumference of the circle (2πr) divided by its radius (r), which equals 2π. The reason focuses only on the circumference of a unit circle without explaining the division by radius.

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Case-Based Questions (CBQ) for Trigonometric Functions

Case Study:

A satellite orbits Earth in a circular path. Engineers need to calculate the satellite's position and speed. The satellite completes one full orbit in 90 minutes and travels along a circular path at a distance of 6,400 km from Earth's centre. To track the satellite, ground stations measure its angular position relative to a fixed point. In one measurement, the satellite sweeps through an arc of 1,500 km along its orbital path. The engineers use both degree and radian measures to ensure precise calculations across different systems.

Q1. What is the angle (in radians) subtended by the arc of 1,500 km at Earth's centre?

  • A. 0.234 radians ✓
  • B. 0.350 radians
  • C. 0.418 radians
  • D. 0.500 radians

Answer: A

Q2. Convert the angle calculated above into degrees.

  • A. 13.4° ✓
  • B. 20.1°
  • C. 23.9°
  • D. 28.6°

Answer: A

Q3. The satellite completes one full orbit (2π radians) in 90 minutes. What is its angular velocity in radians per minute?

  • A. π/90 rad/min
  • B. 2π/90 rad/min
  • C. π/45 rad/min ✓
  • D. 2π/45 rad/min

Answer: C

Q4. If the satellite's position is expressed as an angle from a reference point, and it has rotated through an angle of –225° (negative indicating a different measurement convention), what is this angle in radians?

  • A. –π/4 radians
  • B. –3π/4 radians
  • C. –7π/4 radians
  • D. –5π/4 radians ✓

Answer: D

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