This MCQ module is based on: 3.1 Introduction
TOPIC 8 OF 45
3.1 Introduction
🎓 Class 11
Mathematics
CBSE
Theory
Ch 3 — Trigonometric Functions
⏱ ~15 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: 3.1 Introduction
Targeting Class 11 level in Trigonometry, with Advanced difficulty.
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3.1 Introduction
The word trigonometry comes from the Greek words trigonon (triangle) and metron (measure). It began in ancient times as the geometry of triangles — finding heights, distances and astronomical positions from a few measurable quantities. The Indian astronomer Aryabhata? (476–550 CE) used trigonometric ratios to study planetary motion and shadows, and the modern words "sine" and "cosine" can be traced back to his Sanskrit term jya.
In earlier classes you studied trigonometric ratios for acute angles using right triangles. In this chapter we widen the idea: we let the angle take any real value — positive, negative, or larger than a full turn — and define trigonometric functions of these generalised angles. The natural language for this is the radian, which we develop after first revising the familiar degree? measure.
Aryabhata I
476 – 550 CE
The Indian mathematician–astronomer who, in the Aryabhatiya (499 CE), tabulated values of half-chords (the precursor of sines) for angles in steps of 3.75° and used them to compute eclipses, the rotation of the Earth, and planetary motion. His Sanskrit word jya-ardha ("half chord") evolved through Arabic into the Latin sinus — today's "sine".
3.2 Angles
An angle? is the figure traced when a ray rotates about its endpoint. The starting position of the ray is called the initial side, the final position is the terminal side, and the common endpoint is the vertex.
If the ray rotates anti-clockwise, the angle is taken to be positive; if clockwise, negative. The amount of rotation can be measured — and several measures (revolutions, degrees, radians) all describe the same physical idea.
Fig 3.1: An angle is generated by rotating ray OA to a new position OB.
Fig 3.2: Sense of rotation determines the sign of the angle.
3.2.1 Degree measure
If a complete revolution is divided into 360 equal parts, each part is one degree and is written 1°. So a half-revolution is 180°, a right angle is 90°, and a full turn is 360°. For finer accuracy:
Sub-units of a degree
\(1° = 60'\) (sixty minutes), \(\quad 1' = 60''\) (sixty seconds). Thus \(40°\,30'\) means 40 degrees and 30 minutes, equal to \(40.5°\).
Some often-encountered angles in degree measure are \(360°,\ 180°,\ 270°,\ 420°,\ -30°,\) and \(-420°\). They are illustrated below — the arrow on each diagram shows the rotation that sweeps the initial side onto the terminal side.
Fig 3.3: Six angles in degree measure (positive in yellow, negative in red).
Notice that 420° = 360° + 60°, so its terminal side coincides with that of 60°; similarly −420° has the same terminal side as −60° = 300°. Angles whose terminal sides coincide are called coterminal.
3.2.2 Radian measure
There is a more natural unit of angle for higher mathematics — the radian?. It is defined geometrically using a circle.
Definition: Radian
A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of that circle. In a circle of radius 1 unit, an arc of length 1 unit subtends an angle of 1 radian at the centre.
Fig 3.4: An arc of length \(l\) on a circle of radius \(r\) subtends an angle \(l/r\) radians at the centre.
Because the circumference of a circle of radius \(r\) is \(2\pi r\), the complete angle around the centre is \(\dfrac{2\pi r}{r}=2\pi\) radians. A half-revolution is \(\pi\) radians, and a right angle is \(\dfrac{\pi}{2}\) radians.
Arc length formula
For a circle of radius \(r\), an arc of length \(l\) subtends an angle \(\theta\) (in radians) at the centre where
\[\boxed{\;\theta=\dfrac{l}{r}\quad\Longleftrightarrow\quad l=r\,\theta\;}\]
This is the master formula for converting between arc length and central angle.
3.2.3 Relation between radian and real numbers
Place the unit circle (radius 1, centre at the origin O) on the plane. Let \(A=(1,0)\). Imagine a vertical real number line tangent to the circle at \(A\), with 0 at \(A\) and the positive direction pointing upward.
If we wrap this number line around the circle anti-clockwise, every positive real number \(t\) lands on a unique point \(P_t\) on the circle whose arc from \(A\) has length \(t\). Wrapping clockwise sends every negative real number to a unique point as well. So:
Crucial identification
Every real number corresponds to a unique angle (in radians) at the centre of the unit circle. We can therefore treat radian measures and real numbers as the same thing.
Fig 3.5: Each real number corresponds to a unique point (and hence a unique angle) on the unit circle.
3.2.4 Relation between degree and radian
Since a full revolution is \(2\pi\) radians and also \(360°\), \[2\pi \text{ rad}=360° \quad\Longleftrightarrow\quad \pi\text{ rad}=180°.\]
Conversion formulas
\[\text{Radian measure}=\dfrac{\pi}{180}\times\text{Degree measure},\qquad
\text{Degree measure}=\dfrac{180}{\pi}\times\text{Radian measure}.\]
Using \(\pi\approx \dfrac{22}{7}\) gives the handy approximations
\[1\text{ rad}\approx 57°16',\qquad 1°\approx 0.01746\text{ rad}.\]
The most-used pairs:
| Degree | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|
| Radian | \(\pi/6\) | \(\pi/4\) | \(\pi/3\) | \(\pi/2\) | \(\pi\) | \(3\pi/2\) | \(2\pi\) |
Notation
When an angle is written without any unit, it is understood to be in radians. Thus \(\sin x\) means the sine of \(x\) radians; if degrees are intended we write \(\sin 30°\). For instance, \(\pi=180°\) is shorthand for "\(\pi\) radians equals 180°".
Interactive: Degree ↔ Radian Converter
Drag the slider to change the angle. The arc highlights on the unit circle and both measures update live.
60°
60° = π/3 rad ≈ 1.0472 rad
Worked Examples
Example 1. Convert 40°20′ into radian measure.
We have \(40°20'=40\dfrac{1}{3}°=\dfrac{121}{3}°\). Hence
\[\dfrac{121}{3}°=\dfrac{\pi}{180}\times\dfrac{121}{3}=\dfrac{121\pi}{540}\text{ rad}.\]
Example 2. Convert 6 radians into degree measure.
\(6\text{ rad}=\dfrac{180}{\pi}\times 6\text{ degree}=\dfrac{1080}{\pi}\) degree. Using \(\pi=\dfrac{22}{7}\),
\[6\text{ rad}=\dfrac{1080\times 7}{22}=\dfrac{3780}{11}=343\dfrac{7}{11}\text{ degree}=343°\,38'\frac{2}{11}\text{ minute (approx.)}.\]
Example 3. Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm. (Use \(\pi=22/7\).)
Here \(l=37.4\) cm and \(\theta=60°=60\times\dfrac{\pi}{180}=\dfrac{\pi}{3}\) rad. From \(l=r\theta\),
\[r=\dfrac{l}{\theta}=\dfrac{37.4}{\pi/3}=\dfrac{37.4\times 3}{22/7}=\dfrac{37.4\times 3\times 7}{22}=35.7\text{ cm}.\]
Example 4. The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use \(\pi=3.14\).)
In 60 minutes the minute hand makes one full revolution. So in 40 minutes it turns through \(\dfrac{2}{3}\) revolution \(=\dfrac{2}{3}\times 360°=240°=\dfrac{4\pi}{3}\) rad. Required arc length
\[l=r\theta=1.5\times\dfrac{4\pi}{3}=2\pi\text{ cm}=2\times 3.14=6.28\text{ cm}.\]
Example 5. If arcs of the same length in two circles subtend angles of 65° and 110° at their centres, find the ratio of the radii.
Let \(r_1, r_2\) be the radii. The given angles in radians are
\[\theta_1=65°\times\dfrac{\pi}{180}=\dfrac{13\pi}{36},\qquad \theta_2=110°\times\dfrac{\pi}{180}=\dfrac{22\pi}{36}.\]
Since the arc lengths are equal, \(r_1\theta_1=r_2\theta_2\), so
\[\dfrac{r_1}{r_2}=\dfrac{\theta_2}{\theta_1}=\dfrac{22\pi/36}{13\pi/36}=\dfrac{22}{13}.\]
Hence \(r_1:r_2=22:13\).
Activity: Build a Radian Sector with Paper and String
L3 ApplyMaterials: A paper plate (or a circular cut-out) of any radius \(r\), a length of string equal to \(r\), a protractor, scissors, glue.
Predict: When you bend the string of length \(r\) around the rim, what angle (in degrees) will it subtend at the centre?
- Measure the radius \(r\) of the disc and cut a piece of string of exactly length \(r\).
- Hold one end of the string at a marked point \(A\) on the rim. Lay the string along the rim (it will curve naturally) and mark its other end as point \(B\).
- Draw the radii \(OA\) and \(OB\). The angle \(\angle AOB\) is exactly 1 radian.
- Use a protractor to measure \(\angle AOB\) in degrees.
- Repeat with strings of length \(2r\) and \(\dfrac{r}{2}\). Record the angles measured each time.
A string of length \(r\) gives an angle of about \(57.3°\) (close to \(\dfrac{180°}{\pi}\)). Length \(2r\) gives twice that — \(114.6°\) — and \(\dfrac{r}{2}\) gives \(28.65°\). The angle in degrees is independent of the radius \(r\); it depends only on the ratio \(l/r\). This is exactly the meaning of "radian": angle = arc ÷ radius.
Competency-Based Questions
Scenario: A children's park has a circular merry-go-round of radius 2 m. A child sits on the outer edge. The ride starts from rest and rotates anti-clockwise. After 5 seconds the child has travelled an arc of length 6 m along the rim.
Q1. The angle (in radians) through which the child has turned in 5 seconds is:
L3 ApplyAnswer: (b) 3. Using \(\theta=l/r=6/2=3\) rad. Note this is just under \(\pi\approx3.14\), so the child has gone almost half-way round.
Q2. Convert this angle 3 radians to degrees (use \(\pi=22/7\), give the answer in degrees and minutes).
L3 ApplyAnswer: \(3\text{ rad}=\dfrac{180}{22/7}\times 3=\dfrac{180\times 7\times 3}{22}=\dfrac{3780}{22}=171\dfrac{9}{11}°\). The fractional part is \(\dfrac{9}{11}\) of a degree \(=\dfrac{9}{11}\times 60'\approx 49'\). So \(3\text{ rad}\approx 171°49'\).
Q3. (Fill in the blank) If the merry-go-round of radius 2 m turns through an angle of \(\dfrac{5\pi}{6}\) rad, the arc travelled is _____ m.
L2 UnderstandAnswer: \(l=r\theta=2\times\dfrac{5\pi}{6}=\dfrac{5\pi}{3}\) m \(\approx 5.24\) m.
Q4. (True/False) "Two circles of different radii subtending the same arc length must subtend the same angle at their centres." Justify.
L5 EvaluateFalse. By \(\theta=l/r\), if \(l\) is fixed but \(r\) changes, then \(\theta\) changes inversely. The smaller circle subtends a larger angle for the same arc. (Example 5 in the lesson is exactly this idea reversed.)
Q5. Design: a satellite orbits the Earth (radius of orbit \(R=7000\) km) and sweeps an arc of 1100 km in some time \(\Delta t\). Express the angle traced (i) in radians, (ii) in degrees, and explain why radians are more convenient for the next calculation of angular speed.
L6 CreateSolution: (i) \(\theta=l/R=1100/7000=11/70\approx 0.157\) rad. (ii) \(0.157\times 180/\pi\approx 9°\). Radians are more convenient because angular speed \(\omega=\theta/\Delta t\) is dimensionally \(\text{(unitless)}/s = s^{-1}\) and combines directly with linear speed via \(v=R\omega\). With degrees we would carry an extra \(\pi/180\) factor through every formula.
Assertion–Reason Questions
Assertion (A): The angle 420° has the same terminal side as 60°.
Reason (R): Adding or subtracting any whole-number multiple of 360° to an angle does not change its terminal side.
Reason (R): Adding or subtracting any whole-number multiple of 360° to an angle does not change its terminal side.
Answer: (a). 420° = 360° + 60°, and a full 360° revolution leaves the terminal side unchanged. R is precisely the principle behind A.
Assertion (A): An arc of length \(l\) on a circle of radius \(r\) subtends an angle \(l/r\) radians at the centre.
Reason (R): The radian is defined so that an arc equal in length to the radius subtends 1 radian.
Reason (R): The radian is defined so that an arc equal in length to the radius subtends 1 radian.
Answer: (a). The definition (R) immediately scales: \(k\) times the radius gives \(k\) radians, so an arc of length \(l\) gives \(l/r\) radians.
Assertion (A): 1 radian \(>\) 1 degree.
Reason (R): 1 radian \(\approx\) 57.3°.
Reason (R): 1 radian \(\approx\) 57.3°.
Answer: (a). 1 rad = \(180°/\pi \approx 57.3°\), which is far larger than 1°. R is the numerical reason for A.
Frequently Asked Questions
What is an angle in trigonometry?
An angle is the figure formed by rotating a ray (the initial side) about its endpoint to a new position (the terminal side). The amount of rotation is the measure of the angle and can be positive (anti-clockwise) or negative (clockwise).
How is one degree defined?
If a complete revolution is divided into 360 equal parts, each part is one degree (1°). One degree is further split into 60 minutes (60'), and one minute into 60 seconds (60'').
What is one radian?
One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. The relation l = r·θ (with θ in radians) gives arc length in terms of radius and central angle.
What is the conversion formula between degree and radian?
180° = π radians. Hence Radian measure = (π/180) × Degree measure, and Degree measure = (180/π) × Radian measure.
Why are radians preferred over degrees in calculus?
Radians are dimensionless ratios of arc length to radius, which makes formulas like d/dx(sin x) = cos x clean. With degrees a constant π/180 keeps appearing in derivatives and series expansions.
How do you find arc length given a central angle?
If θ is the central angle in radians and r is the radius, the arc length is l = r·θ. If the angle is given in degrees, first convert to radians using π/180.
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