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Fundamentals of trigonometry: Class 11 Mathematics
Angles in degrees and radians, arc length and sector area, trigonometric ratios and functions, signs in quadrants, values at special angles, fundamental identities, and trigonometric functions of allied angles.
Key points
An angle is formed by rotating a ray about its starting point. The rotation is positive if anticlockwise and negative if clockwise. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x axis. The terminal side then lies in a quadrant, or on an axis, which is called a quadrantal angle.
Angles are measured in degrees or radians. One degree is 1/360 of a full rotation. One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. A full turn is 360° = 2π radians, so 180° = π radians and 1 radian ≈ 57.3°. To convert degrees to radians multiply by π/180. To convert radians to degrees multiply by 180/π.
If a circle has radius r and an arc of length s subtends the angle θ radians at the centre, then s = rθ. The angle must be in radians for this formula. The area of the sector is A = ½r²θ. For example, in a circle of radius 6 cm an angle of π/3 gives an arc of 2π cm.
For a right angled triangle with an acute angle θ, the six trigonometric ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Their reciprocals are cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. Also tan θ = sin θ/cos θ and cot θ = cos θ/sin θ.
For any angle θ, take the point P(x, y) on its terminal side at distance r = √(x² + y²) from the origin. Then sin θ = y/r, cos θ = x/r and tan θ = y/x. These are the trigonometric functions, and they apply to angles of any size, including negative angles. The functions sin and cos are defined for all angles, and their values lie between −1 and 1. The tan function is not defined where cos θ = 0.
Signs of the functions by quadrant:
- Quadrant I (0° to 90°): all ratios are positive.
- Quadrant II (90° to 180°): only sin and cosec are positive.
- Quadrant III (180° to 270°): only tan and cot are positive.
- Quadrant IV (270° to 360°): only cos and sec are positive.
The values at the common angles are worth remembering. For 0°, 30°, 45°, 60°, 90°, sin takes the values 0, 1/2, 1/√2, √3/2, 1. The cos values run in the opposite order: 1, √3/2, 1/√2, 1/2, 0. The tan values are 0, 1/√3, 1, √3, and undefined at 90°.
Fundamental identities (true for all angles where defined):
- sin²θ + cos²θ = 1.
- 1 + tan²θ = sec²θ.
- 1 + cot²θ = cosec²θ.
The identities follow from x² + y² = r². Dividing by r² gives the first. Dividing by x² gives the second, and dividing by y² gives the third. They are used to find the other ratios when one is known, and to simplify or prove expressions. To prove an identity, start from the more complicated side, convert everything to sin and cos, and simplify it to the other side.
Allied angles are angles such as 90° ± θ, 180° ± θ and 360° − θ. Their ratios equal the ratios of θ, up to a sign. Examples: sin(90° − θ) = cos θ, cos(90° − θ) = sin θ, sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ, sin(−θ) = −sin θ and cos(−θ) = cos θ. The sign is decided by the quadrant of the allied angle. The name changes (sin to cos) for 90° and 270° and stays the same for 180° and 360°.
Study tip: Learn the table of sin, cos and tan for 0°, 30°, 45°, 60° and 90°, as many marks depend on it. Always convert degrees to radians before using arc and sector formulas. When proving an identity, work on one side only and change everything to sin and cos.
Definitions and flashcards
Say the meaning first, then open the card to check.
Radian
The angle at the centre of a circle that is subtended by an arc equal in length to the radius.
Standard position
An angle with its vertex at the origin and its initial side on the positive x axis.
Quadrantal angle
An angle whose terminal side lies on an axis, such as 90° or 180°.
Trigonometric ratio
The ratio of two sides of a right angled triangle, such as sin θ.
Identity
An equation that is true for all values of the angle for which both sides are defined.
Allied angles
Angles such as 90° ± θ and 180° ± θ that are related to θ.
Sector
The region of a circle bounded by two radii and the arc between them.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Degrees and radians | π radians = 180° | Convert using π/180 or 180/π. |
| Arc length | s = rθ | θ in radians. |
| Sector area | A = ½r²θ | θ in radians. |
| Basic ratios | sin θ = y/r, cos θ = x/r, tan θ = y/x | r = √(x² + y²). |
| Pythagorean identity | sin²θ + cos²θ = 1 | Always true. |
| Other identities | 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ | Where defined. |
| Co function | sin(90° − θ) = cos θ | Also cos(90° − θ) = sin θ. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define one radian. (2 marks)
One radian is the angle at the centre of a circle subtended by an arc whose length is equal to the radius of the circle. Its size is about 57.3°, and π radians equal 180°.
In which quadrants is sin θ positive? In which is tan θ positive? (2 marks)
sin θ is positive in quadrants I and II, where y is positive. tan θ is positive in quadrants I and III, where x and y have the same sign.
Prove that 1 + tan²θ = sec²θ. (3 marks)
We know sin²θ + cos²θ = 1. Divide every term by cos²θ, for cos θ ≠ 0. This gives sin²θ/cos²θ + 1 = 1/cos²θ. That is tan²θ + 1 = sec²θ.
Find the exact value of sin 150° and cos 120°. (3 marks)
150° = 180° − 30°, which is in quadrant II where sin is positive, so sin 150° = sin 30° = 1/2. 120° = 180° − 60°, which is in quadrant II where cos is negative, so cos 120° = −cos 60° = −1/2.
Derive the formula s = rθ for the length of an arc. (5 marks)
In a circle of radius r, an arc equal to r subtends an angle of 1 radian at the centre. Angle at the centre is proportional to arc length. So an arc of length s subtends an angle θ = s/r radians, because the arc is s/r times as long as the radius. Therefore s = rθ, with θ in radians. For the full circle s = 2πr and θ = 2π, which agrees with the circumference formula.
Show how the sign of each trigonometric ratio is decided in the four quadrants. (5 marks)
Take a point P(x, y) on the terminal side, with r = √(x² + y²) always positive. Then sin θ = y/r has the sign of y, cos θ = x/r has the sign of x, and tan θ = y/x is positive when x and y have the same sign. In quadrant I both x and y are positive, so all are positive. In quadrant II x is negative and y is positive, so only sin (and cosec) is positive. In quadrant III both are negative, so only tan (and cot) is positive. In quadrant IV x is positive and y is negative, so only cos (and sec) is positive.
Numericals with worked steps
Convert 135° to radians and 5π/6 radians to degrees.
- 135° × π/180 = 135π/180.
- Divide top and bottom by 45: 3π/4.
- (5π/6) × 180/π = 5 × 30°.
- = 150°.
Answer: 135° = 3π/4 radians and 5π/6 radians = 150°.
A circle has radius 6 cm. Find the arc length and the sector area for a central angle of 60°.
- θ = 60° × π/180 = π/3 radians.
- s = rθ = 6 × π/3 = 2π ≈ 6.28 cm.
- A = ½r²θ = ½ × 36 × π/3 = 6π.
- 6π ≈ 18.85 cm².
Answer: Arc length 2π ≈ 6.28 cm and sector area 6π ≈ 18.85 cm².
If sin θ = 3/5 and θ is in quadrant II, find cos θ and tan θ.
- cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25.
- So cos θ = ±4/5.
- In quadrant II cos is negative, so cos θ = −4/5.
- tan θ = sin θ/cos θ = (3/5)/(−4/5) = −3/4.
Answer: cos θ = −4/5 and tan θ = −3/4.
Show that (sin θ + cos θ)² + (sin θ − cos θ)² = 2.
- Expand the first: sin²θ + 2 sin θ cos θ + cos²θ.
- Expand the second: sin²θ − 2 sin θ cos θ + cos²θ.
- Add: 2 sin²θ + 2 cos²θ, since the middle terms cancel.
- = 2(sin²θ + cos²θ) = 2(1) = 2.
Answer: The left side equals 2, which proves the identity.
Common mistakes
- Using the formulas s = rθ and A = ½r²θ with θ in degrees. Convert to radians first.
- Forgetting the sign of the ratio in quadrants II, III and IV when finding a value from sin²θ + cos²θ = 1.
- Writing sin²θ as sin θ². It means (sin θ)², the square of the sine.
- Cancelling sin from an expression such as sin θ/sin 2θ as if it were a common factor of the angle.
- Getting the name wrong for allied angles: sin(90° + θ) is cos θ, not sin θ.