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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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Definition of an Angle: An angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side, and the final position of the ray after rotation is called the terminal side of the angle. If the direction of rotation is anticlockwise, the angle is positive; if clockwise, the angle is negative.

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Degree Measure: If a rotation from the initial side to the terminal side is 1360\frac{1}{360}th of a revolution, the angle is said to have a measure of one degree, written as 1∘1^{\circ}. A degree is divided into 60 minutes (1∘=60′1^{\circ} = 60^{\prime}), and a minute is divided into 60 seconds (1′=60′′1^{\prime} = 60^{\prime \prime}). Visually, a complete circle represents a 360∘360^{\circ} rotation.

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Radian Measure: This is the angle subtended at the center of a unit circle (radius 1 unit) by an arc of length 1 unit. In a circle of radius rr, an arc of length ll subtends an angle θ\theta at the center such that θ=lr\theta = \frac{l}{r} radians. Visually, if you wrap a string of length equal to the radius around the edge of the circle, the angle formed at the center is exactly 1 radian.

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Relationship between Degree and Radian: Since a complete revolution subtends an angle of 2π2\pi radians at the center and also measures 360∘360^{\circ}, we establish that 2π radians=360∘2\pi \text{ radians} = 360^{\circ} or π radians=180∘\pi \text{ radians} = 180^{\circ}. This forms the basis for all conversion calculations.

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Arc Length and Sector: In a circle of radius rr, the length of an arc ll that subtends an angle θ\theta (measured in radians) at the center is given by l=rθl = r\theta. Visually, this relationship shows that the arc length is proportional to both the radius and the central angle when the angle is expressed in radians.

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Standard Angle Conversions: Common angles used in trigonometry can be quickly recognized in both systems: 30∘30^{\circ} is π6\frac{\pi}{6}, 45∘45^{\circ} is π4\frac{\pi}{4}, 60∘60^{\circ} is π3\frac{\pi}{3}, 90∘90^{\circ} is π2\frac{\pi}{2}, and 180∘180^{\circ} is π\pi radians.

📐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.