RS Aggarwal Solutions for Class 11 Chapter 14 Measurement Of Angles

Question 1. Using a protractor, draw each of the following angles.
(i) 60° (ii) 130° (iii) 300° (iv) 430°
Answer: The method to draw these angles involves placing the protractor's baseline along the base ray and marking the specified degree measure on the protractor scale, then joining the marked point to the vertex.

For 60°:
Step 1: Draw a line AB.
Step 2: Position the protractor's baseline along BA with the centre at point B.
Step 3: Locate 60° on the protractor scale and mark a small point at the edge, labeling it as P.
Step 4: Connect P to B with a ruler to create the second arm BP of the angle.
Mark the angle with a small arc.

For 130°:
Step 1: Draw a line AB.
Step 2: Position the protractor's baseline along BA with the centre at point B.
Step 3: Locate 130° on the protractor scale and mark a small point at the edge, labeling it as C.
Step 4: Connect C to B with a ruler to create the second arm BC of the angle.
Mark the angle with a small arc.

For 300°:
Step 1: Draw a line AB.
Step 2: Position the protractor's baseline along BA with the centre at point B.
Step 3: Locate 300° on the protractor scale and mark a small point at the edge, labeling it as C.
Step 4: Connect C to B with a ruler to create the second arm BC of the angle.
Mark the angle with a small arc.

For 430°:
Since adding or subtracting 360° does not change an angle's position, 430° can also be expressed as:
430° - 360° = 70°
Therefore, construct an angle of 70° using the same steps as above.
In simple words: Use a protractor to measure the required angle on its scale, mark the point, then draw a line from the vertex through that point to form the angle.

Exam Tip: Always position the protractor's centre exactly at the vertex and align the baseline with one arm; ensure you read the correct scale (inner or outer) matching your baseline direction.

Question 2. Express each of the following angles in radians.
(i) 36° (ii) 120° (iii) 225° (iv) 330° (v) 400° (vi) 7° 30' (vii) -270° (viii) -(22° 30')
Answer: To convert degrees to radians, use the formula: Angle in radians = Angle in degrees × π/180°

(i) 36°
Angle in radians = 36° × π/180° = π/5

(ii) 120°
Angle in radians = 120° × π/180° = 2π/3

(iii) 225°
Angle in radians = 225° × π/180° = 5π/4

(iv) 330°
Angle in radians = 330° × π/180° = 11π/6

(v) 400°
Angle in radians = 400° × π/180° = 20π/9

(vi) 7° 30'
First, convert 30' to degrees: 30'/60 = 0.5°
Total angle = 7° + 0.5° = 7.5°
Angle in radians = 7.5° × π/180° = π/24

(vii) -270°
Angle in radians = -270° × π/180° = -3π/2

(viii) -(22° 30')
First, convert 30' to degrees: 30'/60 = 0.5°
Total angle = 22° + 0.5° = 22.5°
Angle in radians = -22.5° × π/180° = -π/8
In simple words: To change degrees to radians, multiply the degree value by π and divide by 180. This conversion helps in calculus and higher mathematics where radians are the standard unit.

Exam Tip: Always convert minutes to the decimal form of degrees before applying the conversion formula; double-check the sign of negative angles.

Question 3. Express each of the following angles in degrees.
(i) \( \left(\frac{5\pi}{12}\right)^c \) (ii) \( -\left(\frac{18\pi}{5}\right)^c \) (iii) \( \left(\frac{5}{6}\right)^c \) (iv) (-4)^c
Answer: To convert radians to degrees, apply the formula: Angle in degrees = Angle in radians × 180°/π

(i) \( \frac{5\pi}{12} \) radians
Angle in degrees = \( \frac{5\pi}{12} \times \frac{180}{\pi} = 75° \)

(ii) \( -\frac{18\pi}{5} \) radians
Angle in degrees = \( -\frac{18\pi}{5} \times \frac{180}{\pi} = -648° \)

(iii) \( \frac{5}{6} \) radians
Angle in degrees = \( \frac{5}{6} \times \frac{180}{\pi} = 47.7272° \)

To express as degrees, minutes, and seconds:
The decimal part 0.7272° = 0.7272 × 60' = 43.632'
The seconds = 0.632 × 60" = 37.92" or 38"
Final angle = 47° 43' 38"

(iv) -4 radians
Angle in degrees = \( -4 \times \frac{180}{\pi} = -229.0909° \)

To express as degrees and minutes:
The decimal part 0.0909° = 0.0909 × 60' = 5.4545'
The seconds = 0.4545 × 60" = 27.27" or 27"
Final angle = -(229° 5' 27")
In simple words: Multiply the radian measure by 180 and divide by π to get degrees. If you get a decimal, convert the fractional part to minutes and seconds by multiplying by 60 repeatedly.

Exam Tip: Always keep extra decimal places during intermediate steps to ensure accuracy in the final conversion; watch the negative signs carefully.

Question 4. The angles of a triangle are in AP, and the greatest angle is double the least. Find all the angles in degrees and radians.
Answer: Let the three angles in arithmetic progression be represented as: a - d, a, and a + d.

Given that the largest angle equals twice the smallest angle:
\[ a + d = 2(a - d) \]
\[ a + d = 2a - 2d \]
\[ a = 3d \quad \text{...(1)} \]

By the angle sum property of a triangle, the sum of all three angles equals 180°:
\[ (a - d) + a + (a + d) = 180° \]
\[ 3a = 180° \]
\[ a = 60° \quad \text{...(2)} \]

From equations (1) and (2):
\[ 3d = 60° \]
\[ d = 20° \]

Therefore, the three angles are:
\[ a - d = 60° - 20° = 40° \]
\[ a = 60° \]
\[ a + d = 60° + 20° = 80° \]

The required angles are 40°, 60°, and 80°.
In simple words: Set up the angles as terms in an arithmetic sequence, use the condition that the largest is double the smallest to find the common difference, then use the triangle angle sum to find all three angles.

Exam Tip: Always verify that your angles sum to 180° and that the largest is indeed double the smallest before finalizing the answer.

Question 5. The difference between the two acute angles of a right triangle is (π/5)^c. Find these angles in radians and degrees.
Answer: First, convert \( \frac{\pi}{5} \) radians to degrees:
Angle in degree = \( \frac{\pi}{5} \times \frac{180}{\pi} = 36° \)

Let x and y be the two acute angles of the right triangle.

Given condition:
\[ x - y = 36° \quad \text{...(1)} \]

In a right triangle, the sum of the two acute angles is 90°:
\[ x + y = 90° \quad \text{...(2)} \]

Solving equations (1) and (2):
Adding both equations:
\[ 2x = 126° \]
\[ x = 63° \]

Substituting in equation (2):
\[ 63° + y = 90° \]
\[ y = 27° \]

Therefore, the two acute angles are 63° and 27°.

Converting to radians:
For 63°:
\[ \text{Angle in radians} = 63° \times \frac{\pi}{180°} = \frac{7\pi}{20} \]

For 27°:
\[ \text{Angle in radians} = 27° \times \frac{\pi}{180°} = \frac{3\pi}{20} \]
In simple words: Use the two conditions - the difference given and the fact that acute angles in a right triangle sum to 90° - to set up a pair of equations. Solve by adding or substituting to find both angles.

Exam Tip: Always verify that the two acute angles sum to 90° and their difference matches the given value before converting to radians.

Question 6. Find the radius of a circle in which a central angle of 45° intercepts an arc of length 33 cm. (Take π = 22/7)
Answer: We know that the relationship between central angle, arc length, and radius is given by:
\[ \text{Central angle } (\theta) = \frac{\text{arc length}}{radius} \quad \text{...(1)} \]

First, convert the central angle from degrees to radians:
\[ \text{Angle in radians} = 45° \times \frac{\pi}{180°} = \frac{\pi}{4} \]

From equation (1), we can rearrange to find the radius:
\[ \text{Radius} = \frac{\text{arc length}}{\text{central angle}} \]
\[ \text{Radius} = \frac{33}{\pi/4} \]
\[ \text{Radius} = \frac{33 \times 4}{\pi} \]
\[ \text{Radius} = \frac{132}{22/7} \]
\[ \text{Radius} = \frac{132 \times 7}{22} = 42 \]

Therefore, the radius is 42 cm.
In simple words: The central angle equals the arc length divided by the radius. Rearrange this relationship to solve for the radius by dividing the arc length by the angle (in radians).

Exam Tip: Always convert the central angle to radians before using the arc length formula; forgetting this step is a common error leading to incorrect answers.