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Trigonometric Functions - Angles: Degree and Radian Measure

Grade 11CBSE

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

🔑Concepts

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An angle is formed by the rotation of a ray from an initial side to a terminal side. Rotation in the anti-clockwise direction is considered positive, while clockwise rotation is negative.

Diagram showing the initial side, terminal side, and positive rotation of an angle.
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The Radian Measure of an angle is defined by the ratio of the arc length ll to the radius rr of the circle, given by θ=lr\theta = \frac{l}{r}. One radian is the angle subtended at the center of a circle by an arc of length equal to the radius.

A circle showing the relationship between radius, arc length, and the central angle theta in radians.
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In the Degree System, a complete revolution is divided into 360360 equal parts called degrees (∘^{\circ}). Further divisions include minutes (1∘=60′1^{\circ} = 60^{\prime}) and seconds (1′=60′′1^{\prime} = 60^{\prime \prime}).

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To convert degrees to radians, multiply by π180\frac{\pi}{180}. To convert radians to degrees, multiply by 180π\frac{180}{\pi}. Note that π\pi radians corresponds to 180∘180^{\circ}.

📐Formulae

1∘=60′1^{\circ} = 60^{\prime} (1 degree = 60 minutes)

1′=60′′1^{\prime} = 60^{\prime \prime} (1 minute = 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}

θ=lr\theta = \frac{l}{r} where θ\theta is the angle in radians, ll is arc length, and rr is radius

1 radian=180∘π≈57∘16′1 \text{ radian} = \frac{180^{\circ}}{\pi} \approx 57^{\circ} 16^{\prime}

1∘=π180 radians≈0.01746 radians1^{\circ} = \frac{\pi}{180} \text{ radians} \approx 0.01746 \text{ radians}

💡Examples

Problem 1:

Convert 40∘20′40^{\circ} 20^{\prime} into radian measure.

Solution:

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

Explanation:

To convert a degree measure containing minutes or seconds, first convert the entire expression into a decimal or fractional degree before applying the conversion factor π180\frac{\pi}{180}.

Problem 2:

Find the radius of a circle in which a central angle of 60∘60^{\circ} intercepts an arc of length 37.437.4 cm (Use π=227\pi = \frac{22}{7}).

Solution:

Step 1: Convert the angle from degrees to radians. θ=60∘=60×π180=π3\theta = 60^{\circ} = 60 \times \frac{\pi}{180} = \frac{\pi}{3} radians. Step 2: Use the formula l=rθl = r\theta or r=lθr = \frac{l}{\theta}. Step 3: Substitute the given values: l=37.4l = 37.4 and θ=π3\theta = \frac{\pi}{3}. r=37.4π3=37.4×3πr = \frac{37.4}{\frac{\pi}{3}} = \frac{37.4 \times 3}{\pi}. Step 4: Substitute π=227\pi = \frac{22}{7}: r=37.4×3×722=785.422=35.7r = \frac{37.4 \times 3 \times 7}{22} = \frac{785.4}{22} = 35.7 cm.

Explanation:

When using the arc length formula l=rθl = r\theta, the angle θ\theta must always be in radians. If given in degrees, the conversion step is mandatory before calculation.

Problem 3:

Find the degree measure corresponding to 1116\frac{11}{16} radians (Use π=227\pi = \frac{22}{7}).

An angle measuring approximately 39.37 degrees representing 11/16 radians.

Solution:

Degree measure=180π×Radian measure\text{Degree measure} = \frac{180}{\pi} \times \text{Radian measure} Degree measure=180227×1116\text{Degree measure} = \frac{180}{\frac{22}{7}} \times \frac{11}{16} Degree measure=180×7×1122×16=3158 degrees\text{Degree measure} = \frac{180 \times 7 \times 11}{22 \times 16} = \frac{315}{8} \text{ degrees} 3158=3938 degrees\frac{315}{8} = 39 \frac{3}{8} \text{ degrees} 39∘+(38×60)′=39∘22.5′39^{\circ} + \left(\frac{3}{8} \times 60\right)^{\prime} = 39^{\circ} 22.5^{\prime} 39∘22′+(0.5×60)′′=39∘22′30′′39^{\circ} 22^{\prime} + (0.5 \times 60)^{\prime \prime} = 39^{\circ} 22^{\prime} 30^{\prime \prime} Final Answer: 39∘22′30′′39^{\circ} 22^{\prime} 30^{\prime \prime}.

Explanation:

We use the conversion formula from radians to degrees. After getting the fractional degree, we convert the remainder into minutes and then into seconds by multiplying by 60 at each step.

Problem 4:

The minute hand of a clock is 1.51.5 cm long. How far does its tip move in 4040 minutes? (Use π=3.14\pi = 3.14)

A clock face showing the minute hand moving from 12 to 8, covering 40 minutes and forming an arc.

Solution:

  1. In 6060 minutes, the minute hand completes one full revolution, which is 360∘360^{\circ} or 2π2\pi radians.
  2. Angle swept in 4040 minutes, θ=4060×2π=23×2π=4π3\theta = \frac{40}{60} \times 2\pi = \frac{2}{3} \times 2\pi = \frac{4\pi}{3} radians.
  3. The length of the minute hand is the radius, r=1.5r = 1.5 cm.
  4. The distance moved by the tip is the arc length ll.
  5. Using the formula l=rθl = r\theta: l=1.5×4π3l = 1.5 \times \frac{4\pi}{3} l=0.5×4π=2πl = 0.5 \times 4\pi = 2\pi
  6. Substituting π=3.14\pi = 3.14: l=2×3.14=6.28 cml = 2 \times 3.14 = 6.28 \text{ cm}

The tip of the minute hand moves 6.286.28 cm.

Explanation:

To find the distance moved by the tip, we treat the movement as an arc on a circle where the minute hand is the radius. We first find the angle covered in radians (since the formula l=rθl=r\theta requires radians) and then calculate the arc length.