Answer :
Let's solve the given questions step by step:
### Part (a)
If two angles of a triangle are [tex]\(45^\circ\)[/tex] and [tex]\(\left(\frac{\pi}{6}\right)^c\)[/tex], we need to find the remaining angle in degrees.
1. Identify the given angles:
- First angle [tex]\(\angle A = 45^\circ\)[/tex].
- Second angle [tex]\(\angle B = \left(\frac{\pi}{6}\right)^c\)[/tex].
2. Simplify the exponent:
- Assume [tex]\(c = 1\)[/tex] (for simplicity in the given context).
- Then, [tex]\(\angle B = \left(\frac{\pi}{6}\right)^1 = \frac{\pi}{6}\)[/tex].
3. Convert [tex]\(\angle B\)[/tex] to degrees:
- We know that [tex]\(1 \text{ radian} = \frac{180}{\pi}\)[/tex] degrees.
- So, [tex]\(\frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ\)[/tex].
4. Find the remaining angle:
- The sum of angles in a triangle is [tex]\(180^\circ\)[/tex].
- Therefore, the remaining angle [tex]\(\angle C = 180^\circ - 45^\circ - 30^\circ = 105^\circ\)[/tex].
So, the remaining angle is 105°.
### Part (b)
If one angle of a right-angled triangle is [tex]\(25^\circ\)[/tex], find the remaining angle in radian measure.
1. Identify the given angles:
- One right angle, [tex]\(\angle A = 90^\circ\)[/tex].
- Second angle [tex]\(\angle B = 25^\circ\)[/tex].
2. Find the remaining angle:
- The remaining angle in a right-angled triangle [tex]\(\angle C = 90^\circ - 25^\circ = 65^\circ\)[/tex].
3. Convert the remaining angle to radians:
- [tex]\(\angle C \)[/tex] in radians is [tex]\(65^\circ \times \frac{\pi}{180} = \frac{65 \pi}{180} = \frac{13 \pi}{36}\)[/tex].
So, the remaining angle in radian measure is approximately 1.134 rad.
### Part (c)
Two acute angles of a right-angled triangle are [tex]\(63^\circ\)[/tex] and [tex]\(30^3\)[/tex]. Express all angles in radian.
1. Identify the given angles:
- Acute angle [tex]\(\angle A = 63^\circ\)[/tex].
- Acute angle [tex]\(\angle B = 30^3 = 30^3\)[/tex].
2. Actual computation for second acute angle:
- [tex]\(30^3 = 27000^\circ\)[/tex].
3. Right angle:
- The right angle [tex]\(\angle C = 90^\circ\)[/tex] in the triangle.
4. Convert all angles to radians:
- [tex]\(\angle A = 63^\circ \times \frac{\pi}{180} = \frac{63 \pi}{180} = \frac{7 \pi}{20}\)[/tex] approx [tex]\(1.1 \text{ radians}\)[/tex].
- [tex]\(\angle B = 27000^\circ \times \frac{\pi}{180} = 150 \pi \approx 471.2389 \text{ radians}\)[/tex].
- [tex]\(\angle C = 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \approx 1.57 \text{ radians}\)[/tex].
So, the angles in radians are approximately [tex]\((1.1, 471.2389, 1.57)\)[/tex].
### Proofs
#### Proof 1:
If [tex]\(D\)[/tex] and [tex]\(G\)[/tex] are the number of degrees and grades of the same angle, then [tex]\(\frac{G}{10} = \frac{D}{9}\)[/tex].
1. Relation between degrees and grades:
- [tex]\(1 \text{ grade} = \frac{9}{10} \text{ degrees}\)[/tex].
2. Express [tex]\(G\)[/tex] in terms of [tex]\(D\)[/tex]:
- [tex]\(G = \frac{10D}{9}\)[/tex].
3. Divide both sides by 10:
- [tex]\(\frac{G}{10} = \frac{10D}{9} \divide 10 = \frac{D}{9}\)[/tex].
Hence, [tex]\(\frac{G}{10} = \frac{D}{9}\)[/tex] is proved.
#### Proof 2:
If [tex]\(M\)[/tex] and [tex]\(m\)[/tex] represent the number of sexagesimal and centesimal minutes of any angle respectively, prove that [tex]\(\frac{M}{27} = \frac{m}{50}\)[/tex].
1. Define the relations:
- [tex]\(1 \text{ sexagesimal minute} = \frac{1}{60} \text{ degrees}\)[/tex].
- [tex]\(1 \text{ centesimal minute} = \frac{1}{100} \text{ grades} = \frac{1}{100} \times \frac{9}{10} \text{ degrees} = 0.009 \text{ degrees}\)[/tex].
2. Express [tex]\(M\)[/tex] in terms of [tex]\(m\)[/tex]:
- [tex]\(M = m \times 0.54\)[/tex].
3. Divide both sides appropriately:
- [tex]\(\frac{M}{27} = \frac{m \times 0.54}{27} = \frac{m \times 0.54 \divide 0.54}{27 \divide 0.54} = \frac{m}{50}\)[/tex].
Hence, [tex]\(\frac{M}{27} = \frac{m}{50}\)[/tex] is proved.
### Part (a)
If two angles of a triangle are [tex]\(45^\circ\)[/tex] and [tex]\(\left(\frac{\pi}{6}\right)^c\)[/tex], we need to find the remaining angle in degrees.
1. Identify the given angles:
- First angle [tex]\(\angle A = 45^\circ\)[/tex].
- Second angle [tex]\(\angle B = \left(\frac{\pi}{6}\right)^c\)[/tex].
2. Simplify the exponent:
- Assume [tex]\(c = 1\)[/tex] (for simplicity in the given context).
- Then, [tex]\(\angle B = \left(\frac{\pi}{6}\right)^1 = \frac{\pi}{6}\)[/tex].
3. Convert [tex]\(\angle B\)[/tex] to degrees:
- We know that [tex]\(1 \text{ radian} = \frac{180}{\pi}\)[/tex] degrees.
- So, [tex]\(\frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ\)[/tex].
4. Find the remaining angle:
- The sum of angles in a triangle is [tex]\(180^\circ\)[/tex].
- Therefore, the remaining angle [tex]\(\angle C = 180^\circ - 45^\circ - 30^\circ = 105^\circ\)[/tex].
So, the remaining angle is 105°.
### Part (b)
If one angle of a right-angled triangle is [tex]\(25^\circ\)[/tex], find the remaining angle in radian measure.
1. Identify the given angles:
- One right angle, [tex]\(\angle A = 90^\circ\)[/tex].
- Second angle [tex]\(\angle B = 25^\circ\)[/tex].
2. Find the remaining angle:
- The remaining angle in a right-angled triangle [tex]\(\angle C = 90^\circ - 25^\circ = 65^\circ\)[/tex].
3. Convert the remaining angle to radians:
- [tex]\(\angle C \)[/tex] in radians is [tex]\(65^\circ \times \frac{\pi}{180} = \frac{65 \pi}{180} = \frac{13 \pi}{36}\)[/tex].
So, the remaining angle in radian measure is approximately 1.134 rad.
### Part (c)
Two acute angles of a right-angled triangle are [tex]\(63^\circ\)[/tex] and [tex]\(30^3\)[/tex]. Express all angles in radian.
1. Identify the given angles:
- Acute angle [tex]\(\angle A = 63^\circ\)[/tex].
- Acute angle [tex]\(\angle B = 30^3 = 30^3\)[/tex].
2. Actual computation for second acute angle:
- [tex]\(30^3 = 27000^\circ\)[/tex].
3. Right angle:
- The right angle [tex]\(\angle C = 90^\circ\)[/tex] in the triangle.
4. Convert all angles to radians:
- [tex]\(\angle A = 63^\circ \times \frac{\pi}{180} = \frac{63 \pi}{180} = \frac{7 \pi}{20}\)[/tex] approx [tex]\(1.1 \text{ radians}\)[/tex].
- [tex]\(\angle B = 27000^\circ \times \frac{\pi}{180} = 150 \pi \approx 471.2389 \text{ radians}\)[/tex].
- [tex]\(\angle C = 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \approx 1.57 \text{ radians}\)[/tex].
So, the angles in radians are approximately [tex]\((1.1, 471.2389, 1.57)\)[/tex].
### Proofs
#### Proof 1:
If [tex]\(D\)[/tex] and [tex]\(G\)[/tex] are the number of degrees and grades of the same angle, then [tex]\(\frac{G}{10} = \frac{D}{9}\)[/tex].
1. Relation between degrees and grades:
- [tex]\(1 \text{ grade} = \frac{9}{10} \text{ degrees}\)[/tex].
2. Express [tex]\(G\)[/tex] in terms of [tex]\(D\)[/tex]:
- [tex]\(G = \frac{10D}{9}\)[/tex].
3. Divide both sides by 10:
- [tex]\(\frac{G}{10} = \frac{10D}{9} \divide 10 = \frac{D}{9}\)[/tex].
Hence, [tex]\(\frac{G}{10} = \frac{D}{9}\)[/tex] is proved.
#### Proof 2:
If [tex]\(M\)[/tex] and [tex]\(m\)[/tex] represent the number of sexagesimal and centesimal minutes of any angle respectively, prove that [tex]\(\frac{M}{27} = \frac{m}{50}\)[/tex].
1. Define the relations:
- [tex]\(1 \text{ sexagesimal minute} = \frac{1}{60} \text{ degrees}\)[/tex].
- [tex]\(1 \text{ centesimal minute} = \frac{1}{100} \text{ grades} = \frac{1}{100} \times \frac{9}{10} \text{ degrees} = 0.009 \text{ degrees}\)[/tex].
2. Express [tex]\(M\)[/tex] in terms of [tex]\(m\)[/tex]:
- [tex]\(M = m \times 0.54\)[/tex].
3. Divide both sides appropriately:
- [tex]\(\frac{M}{27} = \frac{m \times 0.54}{27} = \frac{m \times 0.54 \divide 0.54}{27 \divide 0.54} = \frac{m}{50}\)[/tex].
Hence, [tex]\(\frac{M}{27} = \frac{m}{50}\)[/tex] is proved.
