Exercise 3.1 answers: Trigonometric Functions

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(i) 25°

1. 25° in radians = 25 × π/180.
2. Cancel the common factor 5: = 5π/36.

Answer5π/36 rad

(ii) –47°30′

1. 30′ = ½°, so –47°30′ = –47.5° = –95/2°.
2. Multiply by π/180: (–95/2) × π/180 = –19π/72.

Answer–19π/72 rad

(iii) 240°

1. 240° in radians = 240 × π/180.
2. Simplify: = 4π/3.

Answer4π/3 rad

(iv) 520°

1. 520° in radians = 520 × π/180.
2. Simplify: = 26π/9.

Answer26π/9 rad

Watch this explained “The pile that only pays”, 12:27 into The exchange rate between the two units, and the arc-length rule it buys

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(i) 11/16

1. Degrees = radians × 180/π.
2. With π = 22/7, 180/π = 1260/22 = 630/11.
3. 11/16 × 630/11 = 630/16 = 315/8.

Answer315/8° (= 39°22′30″)

(ii) –4

1. Degrees = –4 × 630/11 = –2520/11.

Answer–2520/11° (≈ –229°5′27″)

(iii) 5π/3

1. 5π/3 is already a multiple of π, so π cancels: 5π/3 × 180/π = 5 × 60 = 300.

Answer300°

(iv) 7π/6

1. 7π/6 × 180/π = 7 × 30 = 210.

Answer210°

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1. One revolution is a full turn = 2π radians.
2. 360 revolutions in 60 seconds = 360 × 2π radians in 60 seconds.
3. In one second: (720π)/60 = 12π radians.

Answer12π radians

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1. Angle in radians = arc length ÷ radius = 22/100.
2. Convert to degrees: (22/100) × 180/π, using π = 22/7.
3. = (22/100) × (180 × 7/22) = 1260/100 = 12.6° = 63/5°.

Answer63/5° = 12°36′

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1. Radius = diameter ÷ 2 = 20 cm, which is the same as the chord length (20 cm).
2. So the triangle made by the two radii and the chord has all three sides equal — it is equilateral.
3. Every angle of an equilateral triangle is 60°, so the angle at the centre is 60° = π/3 radians.
4. Arc length = radius × angle (in radians) = 20 × π/3.

Answer20π/3 cm

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1. Let the radii be r₁ and r₂, and let l be the common arc length.
2. l = r₁θ₁ = r₂θ₂, so r₁/r₂ = θ₂/θ₁.
3. The unit of the angles cancels in the ratio, so r₁/r₂ = 75/60 = 5/4.

Answerr₁ : r₂ = 5 : 4

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(i) 10 cm

1. θ = arc ÷ radius = 10/75 = 2/15.

Answer2/15 rad

(ii) 15 cm

1. θ = 15/75 = 1/5.

Answer1/5 rad

(iii) 21 cm

1. θ = 21/75 = 7/25.

Answer7/25 rad

Watch this explained “The pile that collects”, 14:01 into The exchange rate between the two units, and the arc-length rule it buys