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Trigonometry - Angles, Degrees and Radian Measure

Grade 11ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Sexagesimal System (Degree Measure) divides one complete revolution into 360360 equal parts called degrees. Each degree (1∘1^\circ) is divided into 6060 minutes (60′60'), and each minute is divided into 6060 seconds (60′′60'').

A circle showing an angle in degrees relative to the horizontal axis.
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The Circular System (Radian Measure) defines 11 radian as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. This leads to the fundamental relationship: s=rθs = r\theta, where θ\theta is in radians.

A sector of a circle where the arc length equals the radius, subtending 1 radian.
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Angle Measurement Direction: An angle is considered positive if it is measured in the counter-clockwise direction from the initial side, and negative if it is measured in the clockwise direction.

Diagram showing positive (CCW) and negative (CW) angle directions.
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The relationship between Radian and Degree measure is based on the semi-circle: 180∘=π radians180^\circ = \pi \text{ radians}. To convert from degrees to radians, multiply by π180\frac{\pi}{180}. To convert from radians to degrees, multiply by 180π\frac{180}{\pi}.

📐Formulae

π radians=180∘\pi \text{ radians} = 180^\circ

1∘=60′1^\circ = 60' (minutes)

1′=60′′1' = 60'' (seconds)

Radian Measure=π180×Degree Measure\text{Radian Measure} = \frac{\pi}{180} \times \text{Degree Measure}

Degree Measure=180π×Radian Measure\text{Degree Measure} = \frac{180}{\pi} \times \text{Radian Measure}

s=rθs = r\theta (where θ\theta is in radians)

Area of Sector=12r2θ\text{Area of Sector} = \frac{1}{2}r^2\theta (where θ\theta is in radians)

💡Examples

Problem 1:

Convert 40∘20′40^\circ 20' into radian measure.

Solution:

Step 1: Convert the minutes into degrees. Since 60′=1∘60' = 1^\circ, then 20′=(2060)∘=(13)∘20' = (\frac{20}{60})^\circ = (\frac{1}{3})^\circ. Step 2: Express the total angle in degrees: 40∘20′=(40+13)∘=(1213)∘40^\circ 20' = (40 + \frac{1}{3})^\circ = (\frac{121}{3})^\circ. Step 3: Convert degrees to radians using the formula Radian=Degree×π180\text{Radian} = \text{Degree} \times \frac{\pi}{180}. Step 4: Radian Measure =1213×π180=121π540= \frac{121}{3} \times \frac{\pi}{180} = \frac{121\pi}{540}.

Explanation:

We first ensure the entire angle is in a single unit (degrees) by converting minutes to a fraction of a degree. Then, we apply the conversion factor π180\frac{\pi}{180} to find the equivalent circular measure.

Problem 2:

Find the length of an arc of a circle of radius 55 cm subtending a central angle measuring 15∘15^\circ.

Solution:

Step 1: Identify the given values: r=5r = 5 cm and θ=15∘\theta = 15^\circ. Step 2: Convert the angle from degrees to radians because the arc length formula s=rθs = r\theta requires θ\theta in radians. θ=15×π180=π12\theta = 15 \times \frac{\pi}{180} = \frac{\pi}{12} radians. Step 3: Substitute the values into the arc length formula: s=5×π12=5π12s = 5 \times \frac{\pi}{12} = \frac{5\pi}{12} cm. Step 4: Using π≈3.14159\pi \approx 3.14159, s≈5×3.1415912≈1.31s \approx \frac{5 \times 3.14159}{12} \approx 1.31 cm.

Explanation:

The most important step in calculating arc length is ensuring the angle is converted to radians. Once the angle is in the correct unit, the arc length is simply the product of the radius and the angle.

Problem 3:

Calculate the distance a point on the rim of a wheel of radius 3535 cm travels when the wheel rotates through an angle of 120∘120^\circ. (Use π=227\pi = \frac{22}{7})

A circle showing a 120 degree sector and a radius of 35 cm.

Solution:

  1. Convert the angle from degrees to radians: θ=120∘×π180=2π3 radians\theta = 120^\circ \times \frac{\pi}{180} = \frac{2\pi}{3} \text{ radians}
  2. Use the arc length formula s=rθs = r\theta: s=35×2π3s = 35 \times \frac{2\pi}{3}
  3. Substitute π=227\pi = \frac{22}{7}: s=35×23×227s = 35 \times \frac{2}{3} \times \frac{22}{7} s=5×443=2203≈73.33 cms = 5 \times \frac{44}{3} = \frac{220}{3} \approx 73.33 \text{ cm} The distance traveled is approximately 73.3373.33 cm.

Explanation:

To find the distance on the rim (arc length), we must first ensure the angle is in radians before applying the formula s=rθs = r\theta.

Problem 4:

Find the angle in degrees subtended at the center of a circle of radius 2121 cm by an arc of length 2222 cm. (Use π=227\pi = \frac{22}{7})

A circle with radius 21 cm and arc length 22 cm, indicating the central angle to be found.

Solution:

  1. Use the formula θ=sr\theta = \frac{s}{r} to find the angle in radians: θ=2221 radians\theta = \frac{22}{21} \text{ radians}
  2. Convert radians to degrees by multiplying by 180π\frac{180}{\pi}: Angle in degrees=2221×180π\text{Angle in degrees} = \frac{22}{21} \times \frac{180}{\pi}
  3. Substitute π=227\pi = \frac{22}{7}: Angle=2221×180×722\text{Angle} = \frac{22}{21} \times \frac{180 \times 7}{22} Angle=13×180=60∘\text{Angle} = \frac{1}{3} \times 180 = 60^\circ The angle is 60∘60^\circ.

Explanation:

The ratio of arc length to radius gives the angle in radians. We then apply the conversion factor to find the value in degrees.