Trigonometry | ISC Class 11 Maths Notes

This note covers angle measurement, radians and degrees, arc length and sector area, the unit circle, trigonometric functions, signs and standard values, domains and ranges, periodicity and graphs, compound angles, multiple angles, half and one-third angles, and transformations between sums and products.

How are positive and negative angles measured?

An angle measures the rotation of a ray about its initial point. The starting ray is the initial side, the final ray is the terminal side, and the point of rotation is the vertex. Anticlockwise rotation gives a positive angle; clockwise rotation gives a negative angle.

An angle records the amount and direction of rotation. Its terminal side alone does not tell us how many complete turns have occurred. A complete revolution is one full rotation, bringing the ray back to its original position.

What do degrees, minutes and seconds mean?

A degree, written °, is one three-hundred-and-sixtieth of a revolution. An angular minute, written ′, is one-sixtieth of a degree. An angular second, written ″, is one-sixtieth of a minute. Thus, 1° = 60′ and 1′ = 60″.

A radian is the angle at the centre of a circle subtended by an arc whose length equals the radius. An arc is part of the circumference, and “subtended” means formed by joining its endpoints to the centre.

A unit circle has radius one unit. Its circumference is 2π units, where π is the ratio of a circle's circumference to its diameter. Consequently, a full revolution is 2π radians, and π radians = 180°.

How do we convert angle measures?

Multiply the degree measure by π/180 to obtain radians. Multiply the radian measure by 180/π to obtain degrees. The angle itself stays the same; only the unit changes. A negative angle keeps its negative sign during conversion.

Note: An angle written without the degree symbol is understood to be in radians. In formulae below, θ, pronounced theta, denotes an angle. The word “radian” is frequently omitted when writing its measure.

Worked example 1. Convert 40°20′ into radians.

Answer: First convert minutes into degrees: 40°20′ = 40 + 20/60 degrees = 121/3 degrees. Then multiply by π/180. The required measure is (121/3)(π/180) = 121π/540 radians. Minutes must be converted before applying the degree-to-radian factor.

How are arc length and sector area calculated?

Let r denote the radius of a circle, S the length of an arc, and θ the positive angle that the arc subtends at the centre, measured in radians. Equal arcs in the same circle subtend equal central angles.

Result: Arc length and circular measure

An arc of length r subtends one radian. An arc of length S therefore subtends S/r radians. Hence θ = S/r and S = rθ. Use the same length unit for S and r before taking their ratio.

The arc length is a distance along the circle. It differs from a chord, which is the straight line segment joining two points on a circle. Substituting a chord length into the arc-length formula would answer a different question.

Worked example 2. A central angle of 60° intercepts an arc of length 37.4 cm. Find the radius, using π = 22/7.

Answer: Convert the angle first: θ = 60π/180 = π/3 radians. Since r = S/θ, the radius is 37.4 ÷ (π/3) = 37.4 × 3 × 7/22 = 35.7 cm. Here cm denotes centimetres.

Worked example 3. A watch's minute hand is 1.5 cm long. Find the distance travelled by its tip in 40 minutes, using π = 3.14.

Answer: The hand completes a revolution in 60 minutes, so the angle turned through is (40/60) × 2π = 4π/3 radians. The tip follows an arc of radius 1.5 cm. Its distance is S = 1.5 × 4π/3 = 2π = 6.28 cm.

How does the sector formula follow?

A sector is the region bounded by two radii and their intercepted arc. Let K denote its area. For a sector with central angle θ between 0 and 2π radians, its fraction of the circle's area is θ/(2π).

Since the circle's area is πr², K = r²θ/2 = rS/2. If d denotes the numerical degree measure instead, K = (d/360)πr². Area has square units; arc length has length units. Keep the angle unit explicit when choosing either formula.

How does the unit circle define the six trigonometric functions?

Place a unit circle with its centre O at the origin, the intersection of the coordinate axes. Let P(a, b) be a point on it: a is its horizontal coordinate and b its vertical coordinate. Let x be the angle from the positive horizontal axis to OP.

The cosine of x, written cos x, is a. The sine of x, written sin x, is b. These definitions extend the ratios of a right-angled triangle to angles beyond acute angles, including negative angles and angles exceeding a complete revolution.

The circle has centre O and axis points A(1, 0), B(0, 1), C(−1, 0) and D(0, −1). Point P(a, b) is in the first quadrant. Its perpendicular meets the horizontal axis at M; OP is labelled 1, OM is a, and MP is b.

See Fig. 3.6 in your NCERT textbook

Identity: The fundamental relation

The right triangle in this construction gives a² + b² = 1. Therefore, sin² x + cos² x = 1 for every real angle x. Here sin² x means (sin x)², the square of the sine value, rather than sin(x²).

A trigonometric identity is an equality valid for all values for which its expressions are defined. The remaining four functions are defined through sine and cosine. A reciprocal is one divided by a non-zero number.

• Tangent: tan x = sin x/cos x, provided cos x is non-zero.
• Cotangent: cot x = cos x/sin x, provided sin x is non-zero.
• Secant: sec x = 1/cos x, provided cos x is non-zero.
• Cosecant: cosec x = 1/sin x, provided sin x is non-zero.

Identity: Two further square relations

Dividing the fundamental identity by cos² x gives 1 + tan² x = sec² x, where cos x is non-zero. Dividing instead by sin² x gives 1 + cot² x = cosec² x, where sin x is non-zero. These restrictions are part of the statements.

How do quadrants determine signs and standard values?

The coordinate axes divide the plane into four regions called quadrants, numbered I, II, III and IV anticlockwise from the upper-right region. The signs of sine and cosine follow the vertical and horizontal coordinates respectively. Quotients and reciprocals then determine the other signs.

Which values should be known exactly?

A quadrantal angle is an integral multiple of π/2, so its terminal side lies on an axis. The sign table concerns points inside quadrants. At an axis, a coordinate can be zero, making a quotient or reciprocal undefined. The symbol √ denotes the non-negative square root.

The radian angles π/6, π/4, π/3 and π/2 correspond to 30°, 45°, 60° and 90° respectively. Obtain reciprocal values only when the corresponding denominator is non-zero. “Not defined” is not another name for zero.

Worked example 4. Given cos x = −3/5 and x in quadrant III, find the other five functions.

Answer: sin² x = 1 − 9/25 = 16/25. Since sine is negative in quadrant III, sin x = −4/5. Therefore cosec x = −5/4 and sec x = −5/3. Finally, tan x = (−4/5)/(−3/5) = 4/3 and cot x = 3/4.

Taking a square root first gives two possible signs, written ±. The quadrant selects the correct one. The numerical magnitude obtained from an identity does not by itself determine the sign of the function.

What are the domains, ranges and periods of these functions?

The domain is the set of permitted inputs; the range is the set of outputs actually attained. Write y for a function's output and n for any integer, meaning a whole number that may be positive, negative or zero.

A periodic function repeats its values after a fixed positive change in its input. Its period here means the smallest positive such change. Adding a full revolution returns a point to the same position on the unit circle, preserving its sine and cosine.

The symbols ≤ and ≥ mean “less than or equal to” and “greater than or equal to”. Sine vanishes at nπ; cosine vanishes at (2n + 1)π/2. These zeros explain the excluded inputs of the quotient and reciprocal functions.

How does periodicity simplify large angles?

The relations sin(x + 2nπ) = sin x and cos(x + 2nπ) = cos x remove complete turns. Tangent and cotangent repeat after π wherever defined. Reduce the angle by an appropriate multiple of the period before evaluating its function.

Worked example 5. Find sin(31π/3).

Answer: Write 31π/3 = 10π + π/3. The term 10π represents five complete turns, so sin(31π/3) = sin(π/3) = √3/2. The input changes by an integral multiple of 2π, while its sine stays the same.

Worked example 6. Find cos(−1710°).

Answer: Add five complete revolutions: −1710° + 5 × 360° = 90°. Cosine is unchanged by this addition, so cos(−1710°) = cos 90° = 0.

How are the six basic trigonometric graphs sketched?

A graph plots each allowed input x horizontally and its output y vertically. Use radians along the horizontal axis. Mark the function's zeros, defined special values, excluded inputs and repeating interval before drawing the curve.

How do sine and cosine vary?

Over one full turn, sine rises from 0 to 1, falls to 0 and then to −1, and rises back to 0. These values occur at 0, π/2, π, 3π/2 and 2π. Cosine takes 1, 0, −1, 0 and 1 at the same inputs.

Both drawings show smooth repeating curves between output levels −1 and 1. The sine curve crosses the origin. The cosine curve passes through output 1 at input 0. Successive waves repeat along the horizontal axis.

See Figs. 3.8 and 3.9 in your NCERT textbook

What happens near an excluded input?

A vertical asymptote is a vertical line approached by a graph as its output becomes arbitrarily large in magnitude. The symbols ∞ and −∞ describe unbounded positive and negative behaviour. They are not attained numerical values of a trigonometric function.

Tangent has vertical asymptotes at odd multiples of π/2 and increases on each continuous branch. Cotangent has them at multiples of π and decreases on each branch. Neither curve should be joined across an excluded input.

The tangent drawing has rising branches and dashed vertical lines at −π/2, π/2 and 3π/2. The cotangent drawing has falling branches with dashed vertical lines at multiples of π. Each drawing repeats after π.

See Figs. 3.10 and 3.11 in your NCERT textbook

How do the reciprocal graphs behave?

Secant is the reciprocal of cosine, so its graph lies at or above 1, or at or below −1. Cosecant is the reciprocal of sine and has the same range. Their asymptotes occur at zeros of their respective denominators.

The secant drawing has an upward branch through (0, 1) and a downward branch through (π, −1). The cosecant drawing has an upward branch through (π/2, 1) and a downward branch through (3π/2, −1).

See Figs. 3.12 and 3.13 in your NCERT textbook

For a sketch, preserve these positions, signs and repeating shapes. In particular, leave the strip between output −1 and output 1 empty for both reciprocal graphs, apart from their boundary points at those output levels.

How are negative and related angles simplified?

Reflecting the unit-circle point for x in the horizontal axis gives the point for −x. Its horizontal coordinate stays unchanged, while its vertical coordinate changes sign. Consequently, cos(−x) = cos x and sin(−x) = −sin x.

An even function has equal values at opposite inputs; an odd function has opposite values at opposite inputs. Cosine and secant are even. Sine, cosecant, tangent and cotangent are odd, with these statements understood on their domains.

Which related-angle identities are useful?

The quarter-turn relations interchange sine and cosine; the half-turn relations preserve the function name and alter signs as shown. These identities apply to real x, not merely to an acute angle used in a drawing.

For tangent, divide the corresponding sine expression by the cosine expression where that denominator is non-zero. For secant and cosecant, take reciprocals where possible. This connects related-angle rules to the definitions instead of treating them as unrelated facts.

For instance, sin(π + x)/cos(π + x) = (−sin x)/(−cos x) = tan x whenever cosine is non-zero. This proves tangent's repetition after π. The signs of numerator and denominator both reverse, leaving the quotient unchanged.

Note: A related-angle identity and a periodicity identity perform different steps. Reflection changes the sign of the angle, whereas adding or removing complete turns changes its measure while preserving its terminal side. Keep track of which operation is being used.

How are compound-angle formulae used?

A compound angle is a sum or difference of angles. In this section, x and y both denote angle measures in the same unit, rather than graph coordinates. A function of their sum is generally not the sum of their function values.

Identity: Sine and cosine of a sum or difference

• sin(x + y) = sin x cos y + cos x sin y.
• sin(x − y) = sin x cos y − cos x sin y.
• cos(x + y) = cos x cos y − sin x sin y.
• cos(x − y) = cos x cos y + sin x sin y.

In the sine formula, the sign between products follows the sign inside the angle. In the cosine formula, it is reversed. Replacing y by −y in either addition identity gives its subtraction form, using the negative-angle identities.

How does the cosine addition proof work?

On a unit circle, compare the chord joining angles x and −y with the chord joining angles x + y and 0. These chords have equal lengths. A chord's squared length is the sum of the squared differences of its endpoints' coordinates.

For the first chord, expansion gives (cos x − cos y)² + (sin x + sin y)² = 2 − 2(cos x cos y − sin x sin y). For the second, it gives 2 − 2cos(x + y). Equating these expressions proves the cosine addition identity.

Worked example 7. Find the exact value of sin 15°.

Answer: Express 15° as 45° − 30°. Then sin 15° = sin 45° cos 30° − cos 45° sin 30° = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2).

This method expresses an unfamiliar angle through familiar standard angles. When the values of the component functions are given instead, first use their quadrants to determine any missing signs, then substitute into the same formula.

How are tangent and cotangent compound angles handled?

Divide the sine addition formula by the cosine addition formula, then divide numerator and denominator by cos x cos y. This gives the tangent formula when the divisions are permitted. The resulting denominator is as important as the numerator.

Identity: Tangent and cotangent formulae

• tan(x + y) = (tan x + tan y)/(1 − tan x tan y).
• tan(x − y) = (tan x − tan y)/(1 + tan x tan y).
• cot(x + y) = (cot x cot y − 1)/(cot y + cot x).
• cot(x − y) = (cot x cot y + 1)/(cot y − cot x).

For the tangent identities, neither component angle nor the resulting sum or difference may be an odd multiple of π/2. For the cotangent identities, none of these angles may be a multiple of π. These conditions ensure every displayed function and denominator is defined.

Worked example 8. Find tan(13π/12).

Answer: Periodicity gives tan(13π/12) = tan(π/12). Since π/12 = π/4 − π/6, use the subtraction formula: (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1) = 2 − √3.

What is the tangent of three added angles?

Let A, B and C denote three angles in the same unit. Write p = tan A, q = tan B and t = tan C. Combining angle-addition formulae gives tan(A + B + C) = (p + q + t − pqt)/(1 − pq − qt − tp).

This form requires cos A, cos B and cos C to be non-zero, and its displayed denominator to be non-zero. It can be obtained by expanding sine and cosine of A + B + C and dividing both by cos A cos B cos C.

If the total angle has zero cosine, its tangent is undefined. A vanishing denominator signals that the quotient cannot be evaluated; it is not permission to assign the answer a value of infinity.

How do double-angle and triple-angle identities follow?

A multiple angle is an integral multiple of another angle. Replacing y by x in the compound-angle formulae produces double-angle relations. Writing 3x as 2x + x then produces triple-angle relations. This gives a connected method of deriving the formulae.

Identity: Double-angle formulae

• sin 2x = 2sin x cos x.
• cos 2x = cos² x − sin² x = 2cos² x − 1 = 1 − 2sin² x.
• tan 2x = 2tan x/(1 − tan² x).

The tangent form requires tan x to exist and 1 − tan² x to be non-zero. Also, sin 2x = 2tan x/(1 + tan² x) and cos 2x = (1 − tan² x)/(1 + tan² x) where tan x exists.

Choose the cosine form that matches the information supplied. If sine is known, 1 − 2sin² x avoids finding cosine. If cosine is known, 2cos² x − 1 avoids finding sine. The original form is useful when both functions already occur.

Identity: Triple-angle formulae

• sin 3x = 3sin x − 4sin³ x.
• cos 3x = 4cos³ x − 3cos x.
• tan 3x = (3tan x − tan³ x)/(1 − 3tan² x).

Here sin³ x and cos³ x mean cubes of the respective function values. The tangent formula requires tan x to exist and 1 − 3tan² x to be non-zero. Sine and cosine forms hold for every real x.

How is the sine triple-angle identity derived?

1. Expand sin(2x + x) as sin 2x cos x + cos 2x sin x.
2. Substitute sin 2x = 2sin x cos x and cos 2x = 1 − 2sin² x.
3. The expression becomes 2sin x cos² x + sin x − 2sin³ x.
4. Replace cos² x by 1 − sin² x and collect terms to obtain 3sin x − 4sin³ x.

The derivation combines angle addition with the fundamental identity. In particular, sin 3x means the sine of the tripled angle, whereas 3sin x means three times a function value. They are different expressions.

How are half-angle and one-third-angle formulae obtained?

A half-angle formula connects the functions of x/2 with those of x. Substitute x/2 for x in the double-angle identities and rearrange. The squared forms are sin²(x/2) = (1 − cos x)/2 and cos²(x/2) = (1 + cos x)/2.

How is the correct square-root sign selected?

Taking square roots gives sin(x/2) = ±√((1 − cos x)/2) and cos(x/2) = ±√((1 + cos x)/2). Each sign is selected from the quadrant of x/2. The quadrant of the original angle cannot simply be reused. In an interval, the symbol < means “less than”.

Where the quotients exist, tan(x/2) = sin x/(1 + cos x) and tan(x/2) = (1 − cos x)/sin x. The second form needs sin x to be non-zero; the first needs 1 + cos x to be non-zero.

Worked example 9. Given tan x = 3/4 and π < x < 3π/2, find sin(x/2), cos(x/2) and tan(x/2).

Answer: sec² x = 1 + 9/16 = 25/16. Since x is in quadrant III, cos x = −4/5. Halving the interval gives π/2 < x/2 < 3π/4, so x/2 lies in quadrant II. Thus sin(x/2) = 3/√10, cos(x/2) = −1/√10 and tan(x/2) = −3.

What does “one-third angle” mean?

A one-third angle is x/3. Set u = x/3, where u denotes the smaller angle. The triple-angle identities become sin x = 3sin u − 4sin³ u, cos x = 4cos³ u − 3cos u, and tan x = (3tan u − tan³ u)/(1 − 3tan² u).

The tangent expression requires tan u to exist and its denominator to be non-zero. Rearranging the sine identity gives 4sin³ u − 3sin u + sin x = 0. These relations produce cubic equations, meaning equations whose highest power of the unknown is three.

The given interval for x determines an interval for x/3 and helps select an appropriate root. One must not replace sin(x/3) by sin x divided by three. Dividing an angle and dividing its function value are different operations.

How do sums become products and products become sums?

Sum-to-product formulae change a sum or difference of two function values into a product. In the following formulae, x and y denote two angles in the same unit. Their half-sum is (x + y)/2 and their half-difference is (x − y)/2.

Identity: Sum-to-product transformations

• sin x + sin y = 2sin((x + y)/2)cos((x − y)/2).
• sin x − sin y = 2cos((x + y)/2)sin((x − y)/2).
• cos x + cos y = 2cos((x + y)/2)cos((x − y)/2).
• cos x − cos y = −2sin((x + y)/2)sin((x − y)/2).

For a cosine difference, keep both the leading negative sign and the order x − y. Reversing one without the other changes the value. These identities follow by adding or subtracting compound-angle formulae and relabelling the two resulting angles.

Worked example 10. Prove that (cos 7x + cos 5x)/(sin 7x − sin 5x) = cot x wherever the original denominator is non-zero.

Answer: The numerator becomes 2cos 6x cos x. The denominator becomes 2cos 6x sin x. The non-zero denominator allows cancellation of 2cos 6x, leaving cos x/sin x = cot x. The original restriction remains part of the identity.

What are the reverse transformations?

• 2cos x cos y = cos(x + y) + cos(x − y).
• −2sin x sin y = cos(x + y) − cos(x − y).
• 2sin x cos y = sin(x + y) + sin(x − y).
• 2cos x sin y = sin(x + y) − sin(x − y).

Adding the cosine compound-angle formulae cancels their sine products and gives the first identity. Subtracting them gives the second. Adding or subtracting the sine compound-angle formulae similarly gives the last two. This explains the signs instead of requiring four disconnected rules.

When proving an identity, transform one side into the other through valid equalities. Before cancelling a factor in a quotient, check that it is non-zero on the original domain. A simplified expression can exist at inputs where the starting quotient does not.

Glossary

• Initial side — The starting position of the ray whose rotation defines an angle.
• Terminal side — The final position of the ray after the stated rotation.
• Radian — The central angle subtended by an arc equal in length to the radius.
• Unit circle — A circle of radius one unit used to define trigonometric functions.
• Sector — The region enclosed by two radii and the arc between their endpoints.
• Quadrant — One of four regions into which coordinate axes divide the plane.
• Quadrantal angle — An integral multiple of π/2 whose terminal side lies on a coordinate axis.
• Domain — The set of input values for which a function is defined.
• Range — The set of output values actually attained by a function.
• Period — The smallest positive input shift after which a periodic function repeats its values.
• Identity — An equality true for every input for which its expressions are defined.
• Vertical asymptote — A vertical line approached as a graph's output becomes unbounded in magnitude.

Common errors and misconceptions

• Misconception: Degree measures can be substituted directly into S = rθ. Correct: This formula requires θ in radians; convert degrees first.
• Misconception: Angular minutes are decimal hundredths of a degree. Correct: There are 60 angular minutes in one degree.
• Misconception: A square root fixes the sign of a trigonometric function. Correct: Use the supplied quadrant to choose the sign after finding the squared value.
• Misconception: An undefined tangent equals infinity. Correct: Division by zero is undefined; infinity describes the behaviour near an excluded input.
• Misconception: sin(x + y) = sin x + sin y. Correct: The compound-angle formula includes products with cosine.
• Misconception: The angle x/2 has the same quadrant as x. Correct: Halve the given interval and identify the new quadrant.
• Misconception: Cancelling factors removes all original domain restrictions. Correct: The original quotient still excludes inputs that make its denominator zero.

Exam-style questions with model answers

Q1. Convert 40°20′ to radians. [2 marks]

1. Convert angular minutes first: 20′ = 20/60 degrees, so 40°20′ = 121/3 degrees.
2. Multiply by π/180 to obtain (121/3)(π/180) = 121π/540 radians.

Q2. A circle has an arc of length 37.4 cm subtending 60° at its centre. Find its radius, using π = 22/7. [3 marks]

1. Convert the central angle into radians before using the arc formula: θ = 60 × π/180 = π/3. Here θ is the central angle in radians.
2. Let S denote the arc length and r the radius. The relation S = rθ gives r = S/θ = 37.4 ÷ (π/3) = 112.2/π cm.
3. Using the specified value π = 22/7 gives r = 112.2 × 7/22 = 35.7 cm. The radius has a length unit, as required.

Q3. Given cos x = −3/5 and x in quadrant III, find sin x, cosec x, sec x, tan x and cot x. [5 marks]

1. Use sin² x + cos² x = 1 to obtain sin² x = 1 − 9/25 = 16/25. Sine is negative in quadrant III, so sin x = −4/5.
2. Cosecant is the reciprocal of sine. Since sine is non-zero, cosec x = 1/(−4/5) = −5/4.
3. Secant is the reciprocal of cosine. The supplied cosine is non-zero, so sec x = 1/(−3/5) = −5/3.
4. Tangent is sine divided by cosine. Therefore tan x = (−4/5)/(−3/5) = 4/3, which is positive in quadrant III.
5. Cotangent is cosine divided by sine. Therefore cot x = (−3/5)/(−4/5) = 3/4, also positive in quadrant III.

Q4. Find the exact value of sin 15° by writing 15° = 45° − 30°. [3 marks]

1. Apply the sine subtraction identity: sin 15° = sin 45° cos 30° − cos 45° sin 30°. The minus sign inside the angle gives subtraction between the two products.
2. Insert the standard values sin 45° = cos 45° = 1/√2, cos 30° = √3/2 and sin 30° = 1/2.
3. Thus sin 15° = √3/(2√2) − 1/(2√2) = (√3 − 1)/(2√2), an exact value without decimal approximation.

Q5. Given tan x = 3/4 and π < x < 3π/2, find sin(x/2), cos(x/2) and tan(x/2). [5 marks]

1. The original angle is in quadrant III. Using sec² x = 1 + tan² x gives sec² x = 25/16, so cos x = −4/5 after selecting the negative sign.
2. Halving the supplied interval gives π/2 < x/2 < 3π/4. Thus the half-angle is in quadrant II, where sine is positive and cosine is negative.
3. Use sin²(x/2) = (1 − cos x)/2 = (1 + 4/5)/2 = 9/10. The quadrant therefore gives sin(x/2) = 3/√10.
4. Use cos²(x/2) = (1 + cos x)/2 = (1 − 4/5)/2 = 1/10. Selecting the negative root gives cos(x/2) = −1/√10.
5. Divide the sine value by the non-zero cosine value: tan(x/2) = (3/√10)/(−1/√10) = −3. Its negative sign agrees with quadrant II.

Q6. Prove (cos 7x + cos 5x)/(sin 7x − sin 5x) = cot x, assuming sin 7x − sin 5x is non-zero. [4 marks]

1. Apply the cosine sum-to-product identity to the numerator. Its half-sum is 6x and half-difference is x, giving cos 7x + cos 5x = 2cos 6x cos x.
2. Apply the sine difference identity to the denominator: sin 7x − sin 5x = 2cos 6x sin x.
3. The question states that this denominator is non-zero, so both cos 6x and sin x are non-zero. Cancelling the common factor 2cos 6x is therefore valid.
4. The quotient reduces to cos x/sin x = cot x, proving the identity throughout the domain specified in the question.

Q7. A circle has radius r and a sector with central angle θ radians, where 0 < θ ≤ 2π. Derive its area K and express it in terms of its arc length S. [3 marks]

1. A complete circle subtends 2π radians at its centre. The sector therefore occupies the fraction θ/(2π) of the circle's area, since both regions have the same radius r.
2. The whole circle has area πr². Hence the sector area is K = [θ/(2π)]πr² = r²θ/2.
3. The arc length is S = rθ, so replacing rθ by S gives K = rS/2. Both formulae express an area in square units.

Key takeaways

• Angles measure directed rotation: anticlockwise is positive, clockwise is negative, and a complete revolution equals 2π radians.
• Convert degrees into radians before using S = rθ or the sector formula K = r²θ/2.
• The unit circle defines cosine as a horizontal coordinate and sine as a vertical coordinate.
• Use the quadrant to select signs, especially after taking square roots in an identity.
• Sine, cosine, secant and cosecant have period 2π; tangent and cotangent have period π.
• Compound-angle identities generate multiple-angle identities and transformations between sums and products of trigonometric functions.
• For half angles, first halve the given interval; for one-third angles, use the triple-angle relations.
• Domain restrictions survive simplification: a cancelled denominator factor cannot make an originally undefined input valid.

Test yourself

What distinguishes positive and negative angles?

Anticlockwise rotation gives a positive angle; clockwise rotation gives a negative angle.

What does one radian mean?

It is the central angle subtended by an arc equal in length to the circle's radius.

Why is tan(π/2) undefined?

Tangent equals sine divided by cosine, and cosine is zero at π/2.

Which functions are positive inside quadrant III?

Tangent and cotangent are positive because sine and cosine are both negative.

What is the range of secant?

Its values satisfy y ≤ −1 or y ≥ 1, where y denotes the function's output.

How does sin 2x differ from sin² x?

The first is sine of twice the angle; the second is the square of its sine value.

Why can a half-angle formula need a negative square root?

The half-angle's quadrant determines the sign, even though its squared function value is non-negative.

How is cos x − cos y written as a product?

It equals −2sin((x + y)/2)sin((x − y)/2), with the leading negative sign retained.