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Lesson 1 of 5

Trigonometric Functions · Lesson 1 of 5

Angles, Degrees and Radians

“Understand angles as rotation and use radians to connect turning, circles and distance travelled.”

Learning Objectives

• Describe positive, negative and multi-revolution angles using their initial and terminal sides. • Convert between degrees, minutes, seconds and radians. • Explain why one radian depends on the ratio of arc length to radius. • Connect real-number inputs to signed movement around a unit circle. • Use arc length to solve circle, clock, wheel and pendulum problems.

From triangles to rotation

Trigonometry first connects the angles and sides of triangles. But a turning wheel, a rotating ray and a repeating wave are not limited to the acute angles inside a right triangle. To describe them, we need angles that can be negative, larger than a full turn, or any real-number amount of rotation.

Applications include navigation, surveying, tides, vibration, musical tones and electrical circuits. The common idea is that a quantity changes in a way related to an angle or a repeating motion. This chapter develops the mathematical language behind that idea; we begin by making the meaning and measurement of an angle precise.

The word trigonometry comes from Greek roots meaning triangle and measurement. Its triangle origins remain useful even as we extend the subject to rotations and functions.

Definition
Angle as rotation

An angle measures the rotation of a ray about its fixed endpoint. The starting ray is its initial side, the final ray is its terminal side, and the fixed endpoint is its vertex.

Anticlockwise rotation is taken as positive and clockwise rotation as negative. The sign records direction; the magnitude records how much turning occurred. A full anticlockwise turn is 360°, while a full clockwise turn is −360°. Both return to the initial ray, but they describe different rotations.

Anticlockwise: positiveClockwise: negativeO+60°O−60°Each starts from the right-facing ray. The terminal sides differ.
Signed rotation— Follow the turn from the initial horizontal ray to the terminal ray. The sign distinguishes the direction.

Angles such as 420° and 60° end on the same ray because 420° = 360° + 60°. Likewise, −420° is one full clockwise turn followed by a further 60° clockwise turn. The terminal ray alone cannot tell you how many complete turns have occurred; a rotation measure retains that information.

Example — Reading a multi-turn angle

Problem
Describe the rotations 270°, 420°, −30° and −420°.

  1. 1.270° is three quarters of an anticlockwise turn. Starting from the positive horizontal ray, it ends on the downward vertical ray.
  2. 2.420° = 360° + 60°: one full anticlockwise turn followed by 60° more. It ends where a 60° turn ends.
  3. 3.−30° is a 30° clockwise turn. −420° = −360° − 60° is one full clockwise turn followed by 60° more clockwise.
  4. 4.Although some rotations share a terminal ray, the signed amounts of rotation are not equal.

Degrees, minutes and seconds

Degree measure divides a full revolution into 360 equal parts. Smaller subdivisions allow us to express angles more precisely without using decimals. These subdivisions are units of angle, even though their names are also used for time.

Definition
Degree

One degree, written 1°, is one three-hundred-and-sixtieth of a complete revolution.

Subdivisions of a degreeLaTeX
′ denotes an angular minute and ″ an angular second. They are not minutes and seconds of elapsed time.

Thus 20′ = 20/60° = 1/3°, and 30″ = 30/3600° = 1/120°. In 40°20′, add the two parts: 40 + 1/3 degrees. Do not read 40°20′ as the decimal 40.20°; decimal hundredths and sixtieths are different fractions.

Example — Using the sixty-based subdivisions

Problem
Express 12°18′30″ in decimal degrees.

  1. 1.Convert each part into degrees: 18′ = 18/60° = 0.3° and 30″ = 30/3600° = 1/120°.
  2. 2.Add the parts: 12 + 0.3 + 1/120 = 12.308333... degrees.
  3. 3.The decimal representation describes the same rotation. Its recurring decimal need not be rounded until a precision is requested.
A minus sign applies to the whole angle

The notation −47°30′ is normally read as −(47° + 30′), or −47.5°. It is not −47° + 0.5°. Stating the brackets during conversion makes the direction unambiguous.

Radians: matching an arc to a radius

Degree measure divides a turn by a chosen number, 360. Radian measure instead compares two lengths in the circle itself: the travelled arc and the radius. This makes radians especially natural whenever turning and distance are connected.

Definition
Radian

One radian is the central angle subtended by an arc whose length equals the radius of the circle. In a unit circle, one radian corresponds to an arc of length one.

If the radius doubles, the arc for the same central angle also doubles. Their ratio stays fixed. That is why radians describe the angle independently of the size of the circle. A radian is not a length; it is an angle measure obtained from a ratio of two lengths measured in the same units.

1 radrarc length = rOarc/radius = 1angle = 1 radian
One radian— The highlighted arc is as long as the radius. Its angle is one radian in a circle of any size.

For two radii of arc length the angle is two radians; for half a radius of arc length it is half a radian. In general, if an arc of length l subtends a central angle θ in radians in a circle of radius r, the angle is l/r. Rearranging gives the distance formula l = rθ.

Radian measure and arc lengthLaTeX
r is the positive radius and l is the arc length. For an ordinary nonnegative travelled arc, θ is the nonnegative angle swept in radians. For a directed angle, the arc-length magnitude is r|θ|.

A full circumference has length 2πr. Dividing it by r gives 2π radians for a full turn. This conclusion does not depend on the radius. In particular, a half turn is π radians and a quarter turn is π/2 radians.

Use radians in l = rθ

If θ is given in degrees, convert it before multiplying by r. Putting 60 directly into l = rθ treats the angle as 60 radians, which is very different from 60°. Also use the radius, not the diameter.

Real numbers can label any amount of turning

A unit circle turns signed arc lengths into angle measures. Imagine laying a number line tangent to the circle at its rightmost point, with zero at that contact point. Wrap its positive part anticlockwise and its negative part clockwise around the circle.

The number 1 measures one unit of anticlockwise arc and therefore one radian. The number −1 measures one unit of clockwise arc and therefore −1 radian. Larger numbers keep winding around the circle. Every real number specifies a signed rotation, but different numbers separated by a whole circumference reach the same terminal point.

0+1−1Positive length wraps anticlockwiseNegative length wraps clockwiseO
Wrapping a number line around a unit circle— Signed distances along the tangent become signed radian rotations. Full extra turns preserve the final point but change the angle measure.

This is the bridge from angles to functions on real-number inputs. In later lessons sin x and cos x will accept any real x, understood as a radian measure. They will depend on the terminal point, so repeated windings will produce repeated values.

Converting between degrees and radians

Both units measure the same rotation, so conversion changes the number used to express the angle rather than changing the angle itself. Use the half-turn equality as the reference: π radians correspond to 180°. Multiplication by the appropriate conversion factor preserves that equality.

Degree–radian conversionLaTeX

When a degree symbol appears, interpret the input as degrees. In this chapter a bare angle such as π/4, 2 or x is measured in radians unless another unit is stated. Thus sin 1 and sin 1° refer to different angles. Keeping the convention visible matters more than memorising a decimal conversion.

Degrees0°30°45°60°90°180°270°360°
Radians0π/6π/4π/3π/2π3π/22π

Exactly, one radian equals 180/π degrees, approximately 57.2958°, and one degree equals π/180 radians, approximately 0.0174533. If a problem prescribes π ≈ 22/7 or 3.14, use that approximation consistently and identify the answer as approximate. An angle written as a fraction of π often converts exactly without any approximation.

Example — Degrees and minutes to radians

Problem
Convert 40°20′ into radians.

  1. 1.First express the entire measure in degrees: 40°20′ = (40 + 20/60)° = (121/3)°.
  2. 2.Multiply by π/180: (121/3)(π/180) = 121π/540 radians.
  3. 3.The answer is exact. The 20 minutes were converted before applying the degree-to-radian factor.
Example — Radians to degrees, minutes and seconds

Problem
Convert 6 radians to degrees, minutes and seconds using π ≈ 22/7.

  1. 1.The degree measure is 6 × 180/(22/7) = 3780/11 = 343 + 7/11 degrees.
  2. 2.Convert the fractional degree: (7/11) × 60 = 420/11 = 38 + 2/11 angular minutes.
  3. 3.Convert the remaining fractional minute: (2/11) × 60 = 120/11 ≈ 10.909 angular seconds.
  4. 4.Using this prescribed approximation, the result is about 343°38′11″. With a more precise π the numerical approximation changes; do not present a 22/7-based answer as exact.

Arc length in moving systems

The same relationship applies whether an arc is marked on a circle or traced by a moving tip. Identify the radius of the path and the angle swept. If time is involved, first convert elapsed time into a fraction or number of revolutions.

Example — Finding a radius

Problem
A central angle of 60° intercepts an arc of length 37.4 cm. Find the radius using π ≈ 22/7.

  1. 1.Convert 60° to π/3 radians. Rearrange l = rθ to get r = l/θ.
  2. 2.Thus r = 37.4/(π/3) = 112.2/π cm. Using π ≈ 22/7 gives r ≈ 112.2 × 7/22 = 35.7 cm.
  3. 3.The arc is slightly longer than the radius because its angle π/3 is slightly more than one radian. This provides a useful reasonableness check.
Example — A clock-hand tip

Problem
A minute hand is 1.5 cm long. How far does its tip move in 40 minutes, using π ≈ 3.14?

  1. 1.A minute hand completes a turn in 60 minutes. In 40 minutes it makes 40/60 = 2/3 of a revolution, sweeping a magnitude of (2/3)(2π) = 4π/3 radians.
  2. 2.The path radius is the hand length 1.5 cm. Arc distance = 1.5 × 4π/3 = 2π cm ≈ 6.28 cm.
  3. 3.The hand turns clockwise, so its directed angle is negative under our convention, but the distance travelled is positive. We used the magnitude of the angle for the distance.
Example — Equal arcs in two circles

Problem
Equal-length arcs subtend 65° and 110° in circles of radii r₁ and r₂. Find r₁:r₂.

  1. 1.Their radian angles are θ₁ = 65π/180 = 13π/36 and θ₂ = 110π/180 = 22π/36.
  2. 2.Because the arc lengths are equal, r₁θ₁ = r₂θ₂. Dividing by r₂θ₁ gives r₁/r₂ = θ₂/θ₁.
  3. 3.Thus r₁/r₂ = (22π/36)/(13π/36) = 22/13, so the ratio is 22:13.
  4. 4.The smaller angle belongs to the larger radius: the same arc length occupies a smaller fraction of a larger circumference.
Example — Turning rate of a wheel

Problem
A wheel makes 360 revolutions per minute. Through how many radians does it turn in one second?

  1. 1.Divide by 60 to get 6 revolutions per second.
  2. 2.Each revolution sweeps 2π radians, so in one second the magnitude of its rotation is 6 × 2π = 12π radians.
  3. 3.This describes total rotation, not just the final ray. After six complete turns the terminal ray returns to its starting position.
Example — From a chord to its minor arc

Problem
A circle has diameter 40 cm and a chord of length 20 cm. Find the length of the minor arc cut off by the chord.

  1. 1.The radius is 20 cm. Join the centre to both endpoints of the chord. Both radii and the chord have length 20 cm, so the triangle is equilateral.
  2. 2.Its central angle is 60° = π/3 radians. The minor arc corresponds to this smaller angle, not to the remaining 300° around the circle.
  3. 3.Arc length = rθ = 20π/3 cm. A straight chord and its curved arc are different lengths; here 20π/3 is slightly larger than 20.

A pendulum tip provides another circular arc: the pendulum length is the radius of its path. If the length is 75 cm and the tip travels an arc of 15 cm, its angular sweep is 15/75 = 1/5 radian. This uses the arc along the path, not the straight distance between the tip’s endpoint positions.

Quiz

Quick check

What does a negative angle describe?

Quick check

Which radian measure equals 240°?

Quick check

What is 40°20′ in decimal degrees?

Quick check

A 12 cm radius sweeps 1.5 radians. What is the arc length?

Quick check

Which statement about one radian is correct?

Quick check

Which pair has the same terminal ray from the same initial side?

Practice Problems

Practice Problems
  1. Convert 25°, −47°30′, 240° and 520° to exact radian measures.
  2. Convert 11/16 radians and −4 radians to degrees using π ≈ 22/7; also convert 5π/3 and 7π/6 exactly.
  3. Explain why a circle of any radius has a full-turn angle of 2π radians.
  4. A radius-100 cm circle has an arc of length 22 cm. Find its central angle in degrees using π ≈ 22/7.
  5. In two circles, equal arcs subtend 60° and 75°. Find the ratio of the first radius to the second, with reasoning.
  6. A 75 cm pendulum tip travels arcs of 10, 15 and 21 cm. Find the three angular sweeps in radians.
  7. A wheel completes 15 turns in one second. Give its total radian sweep and explain why its final ray does not identify the number of turns.
  8. Draw 1 radian and −1 radian on a unit circle. Explain what stays the same and what changes.

Key Takeaways

Key Takeaways

• An angle is a signed amount of rotation, not merely the final position of a ray. • A degree has sixty angular minutes, and an angular minute has sixty angular seconds. • Radian measure compares arc length with radius: θ = l/r and l = rθ for a nonnegative angular sweep. • A full turn is 360° or 2π radians; convert units before using an arc-length formula. • Real numbers specify signed radian rotations, including repeated complete turns. • For moving tips, distinguish distance along an arc from straight endpoint distance and use the radius of the path.

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Trigonometric Functions, the Unit Circle and Graphs