Grade 9: Exercise 2.1 (A) [ Trigonometry ]

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Exercise 2.1 (A) — Angle systems & triangle problems

📖 Sexagesimal ↔ Centesimal conversions | Right angles & ratios

❖ Question 1

Fill in the blanks:

• (a) 1 right angle = \(\ldots\ldots\) (degrees)
  1 right angle = \(90^\circ\)
• (b) 1 right angle = \(\ldots\ldots\) (grades)
  1 right angle = \(100^\text{g}\)
• (c) 2 right angles = \(\ldots\ldots\) (grades)
  2 right angles = \(200^\text{g}\)
• (d) \(200^\text{g} = \ldots\ldots\) (degrees)
  \(200^\text{g} = 180^\circ\)
• (e) \(1^\circ = \left(\frac{\dots}{\dots}\right)^\text{g}\)
  \(1^\circ = \left(\frac{10}{9}\right)^\text{g}\)
• (f) \(1^\text{g} = \left(\frac{\dots}{\dots}\right)^\circ\)
  \(1^\text{g} = \left(\frac{9}{10}\right)^\circ\)

❖ Question 2

Convert to seconds in the sexagesimal system:

• (a) \(35'\)
  \begin{align*} 35' &= (35\times 60 )'' \\ &= 2100'' \end{align*}
• (b) \(50' \, 40''\)
  \(50'\,40'' = (50\times60 + 40)'' = 3040''\)
• (c) \(30^\circ \, 40' \, 50''\)
  \(30^\circ 40' 50'' = (30\times3600 + 40\times60 + 50)'' = 110450''\)
• (d) \(55^\circ \, 30' \, 10''\)
  \(55^\circ 30' 10'' = 199810''\)
• (e) \(10^\circ \, 25' \, 48''\)
  \(10^\circ 25' 48'' = 37548''\)
• (f) \(55^\circ \, 56' \, 28''\)
  \(55^\circ 56' 28'' = 201388''\)

❖ Question 3

Define the sexagesimal system and centesimal system, and explain where they are used. How is conversion done? Write with examples.

Sexagesimal System (Degree System): Base 60. \(1 \text{ full circle}=360^\circ\), \(1^\circ=60'\), \(1'=60''\). Used in geometry, navigation, astronomy, time.
Centesimal System (Grade System): Base 100. \(1 \text{ full circle}=400^\text{g}\), \(1 \text{ right angle}=100^\text{g}\), \(1^\text{g}=100'\), \(1'=100''\). Used in surveying, civil engineering.
Conversion: \(90^\circ = 100^\text{g}\) → \(1^\circ = \left(\frac{10}{9}\right)^\text{g}\), \(1^\text{g} = \left(\frac{9}{10}\right)^\circ\).
Example: \(45^\circ = 45\times\frac{10}{9}=50^\text{g}\); \(80^\text{g}=80\times\frac{9}{10}=72^\circ\).

❖ Question 4

Convert to the sexagesimal system (degrees):

• (a) \(30^\text{g} \, 20' \, 10''\)
  \(30^\text{g}20'10'' = 27^\circ 10' 51.24''\)
• (b) \(25^\text{g} \, 15' \, 10''\)
  \(= 22^\circ 38' 9.24''\)
• (c) \(45^\text{g} \, 35' \, 25''\)
  \(= 40^\circ 49' 2.1''\)
• (d) \(30^\text{g} \, 12'\)
  \(= 27^\circ 6' 28.8''\)
• (e) \(26^\text{g} \, 15''\)
  \(= 23^\circ 24' 4.86''\)
• (f) \(47^\text{g} \, 48' \, 49''\)
  \(= 42^\circ 44' 11.08''\)

❖ Question 5

Convert to the centesimal system (grades):

• (a) \(25^\circ \, 45' \, 30''\) → \(28^\text{g} 62' 3''\)
• (b) \(30^\circ \, 15' \, 15''\) → \(33^\text{g} 61' 57.41''\)
• (c) \(49^\circ \, 50' \, 25''\) → \(55^\text{g} 37' 80.86''\)
• (d) \(44^\circ \, 35' \, 25''\) → \(49^\text{g} 54' 47.53''\)
• (e) \(80^\circ \, 50' \, 20''\) → \(89^\text{g} 82' 10''\)
• (f) \(76^\circ \, 26' \, 33''\) → \(84^\text{g} 93' 61.11''\)

❖ Question 6

Convert to degrees:

\(50^\text{g} = 45^\circ\)

\(80^\text{g} = 72^\circ\)

\(130^\text{g} = 117^\circ\)

\(160^\text{g} = 144^\circ\)

\(70^\text{g} = 63^\circ\)

\(250^\text{g} = 225^\circ\)

❖ Question 7

Convert to grades (decimal/compound form):

• \(50^\text{g}40'8'' = 50.4008^\text{g}\)
• \(40^\text{g}32'33'' = 40.3233^\text{g}\)
• \(56^\text{g}85'50'' = 56.8550^\text{g}\)
• \(45^\text{g}35'' = 45.0035^\text{g}\)
• \(37^\text{g}50' = 37.50^\text{g}\)
• \(98^\text{g}42'37'' = 98.4237^\text{g}\)

❖ Question 8

Convert to grades:

\(45^\circ = 50^\text{g}\)   \(270^\circ = 300^\text{g}\)   \(18^\circ = 20^\text{g}\)
\(36^\circ = 40^\text{g}\)   \(108^\circ = 120^\text{g}\)   \(54^\circ = 60^\text{g}\)

❖ Question 9

One angle of a right angled triangle is \(60^\circ\).

• What should be the measure of one angle of a triangle to be a right angled triangle?
  One angle must be \(90^\circ\).
• In a right-angled triangle, what should be the grade measure of the largest angle?
  \(100^\text{g}\) (since \(90^\circ = 100^\text{g}\)).
• Find the measure of the remaining angle in degrees.
  Third angle = \(90^\circ - 60^\circ = 30^\circ\).

❖ Question 10

The three interior angles of a triangle are in the ratio \(1:2:3\). Find each angle in degrees.

Let angles be \(x,2x,3x\). Sum: \(6x=180^\circ \Rightarrow x=30^\circ\). Angles: \(30^\circ, 60^\circ, 90^\circ\).

❖ Question 11

The ratio of the three angles of a triangle is \(2:3:4\).

• Find measures in degrees → \(40^\circ, 60^\circ, 80^\circ\).
• Find measures in grades → \(44\frac{4}{9}^\text{g},\; 66\frac{2}{3}^\text{g},\; 88\frac{8}{9}^\text{g}\).

Let angles \(2x,3x,4x\): \(9x=180^\circ\) → \(x=20^\circ\). Grades: multiply by \(\frac{10}{9}\).

❖ Question 12

The ratio of the angles of a triangle is \(5:7:8\).

• Degrees: \(45^\circ, 63^\circ, 72^\circ\).
• Grades: \(50^\text{g}, 70^\text{g}, 80^\text{g}\).

\(5x+7x+8x=180^\circ \rightarrow 20x=180 \Rightarrow x=9^\circ\).

❖ Question 13

Quadrilateral interior angles ratio \(3:4:5:6\).

• Measures in degrees: \(60^\circ, 80^\circ, 100^\circ, 120^\circ\).
• Smallest & largest into grades: \(60^\circ = 66\frac{2}{3}^\text{g}\), \(120^\circ = 133\frac{1}{3}^\text{g}\); difference = \(66\frac{2}{3}^\text{g}\).

\(3x+4x+5x+6x=360^\circ \rightarrow 18x=360^\circ \Rightarrow x=20^\circ\).

❖ Question 14

In a triangle, one angle is \(72^\circ\). Remaining two angles ratio \(1:3\).

• Remaining two angles (degrees): \(27^\circ\) and \(81^\circ\).
• All angles in grades: \(80^\text{g}, 30^\text{g}, 90^\text{g}\).
• If ratio \(1:5\) instead → angles: \(18^\circ, 90^\circ\) (triangle: \(72^\circ,18^\circ,90^\circ\)).

\(72+x+3x=180 \Rightarrow 4x=108 \rightarrow x=27^\circ\). For \(1:5\): \(72+y+5y=180 \Rightarrow 6y=108 \rightarrow y=18^\circ\).

❖ Question 15

Two angles of a triangle in ratio \(3:4\) and third angle = \(60^\text{g}\).

• \(60^\text{g}\) in degrees = \(54^\circ\).
• All angles in degrees: \(54^\circ, 72^\circ, 54^\circ\).
• Type: acute isosceles triangle (two equal angles).

\(3x+4x+54=180 \Rightarrow 7x=126 \Rightarrow x=18^\circ\) → angles \(54^\circ,72^\circ,54^\circ\).

❖ Question 16

Right-angled triangle, one acute angle = \(\frac{3}{10}\) of a right angle.

• One right angle = \(90^\circ\).
• \(\frac{3}{10}\) of right angle = \(27^\circ\).
• Other acute angle in grades: \(63^\circ = 70^\text{g}\).

Sum of acute angles = \(90^\circ\) → other acute = \(63^\circ\), convert: \(63\times\frac{10}{9}=70^\text{g}\).

❖ Question 17

Sum of two angles = \(100^\circ\), difference = \(20^\text{g}\).

• \(1^\circ\) in grades = \(\frac{10}{9}^\text{g}\).
• Angles in degrees: \(59^\circ\) and \(41^\circ\).
• Angles in grades: \(65\frac{5}{9}^\text{g}\) and \(45\frac{5}{9}^\text{g}\).

\(20^\text{g}=18^\circ\). Solve \(x+y=100, x-y=18 \rightarrow x=59, y=41\). Grades: multiply by \(10/9\).

❖ Question 18

Sum of two angles = \(45^\circ\), difference = \(30^\text{g}\).

• \(1^\text{g} = 0.9^\circ\) (or \(\frac{9}{10}^\circ\)).
• Angles in degrees: \(36^\circ\) and \(9^\circ\).
• Angles in grades: \(40^\text{g}\) and \(10^\text{g}\).

\(30^\text{g}=27^\circ\). System: \(x+y=45, x-y=27 \rightarrow x=36, y=9\).

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