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First Year Math Application of Trigonometry 7. Prove that in an equilateral trianglei) r: R: r_{1}=1: 2: 3


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7. Prove that in an equilateral trianglei) r: R: r_{1}=1: 2: 3

Q.3 If you are looking a bird in the tree from the ground then the angle form is the(a) angle of elevation(b) angle of depression(c) constant angle(d) right angle
Q.3 If you are looking a bird in the tree from the ground then the angle form is the(a) angle of elevation(b) angle of depression(c) constant angle(d) right angle

Q.3 If you are looking a bird in the tree from the ground then the angle form is the(a) angle of elevation(b) angle of depression(c) constant angle(d) right angle

12. The angle of elevation of the top of a 60 \mathrm{~m} high tower from a point A on the same level as the foot of the tower is 25^{\circ} . Find the angle of elevation of the top of the tower from a point B 20 \mathrm{~m} nearer to A from the foot of the tower.
12. The angle of elevation of the top of a  60 \mathrm{~m}  high tower from a point  A  on the same level as the foot of the tower is  25^{\circ} .  Find the angle of elevation of the top of the tower from a point  B 20 \mathrm{~m}  nearer to  A  from the foot of the tower.

12. The angle of elevation of the top of a 60 \mathrm{~m} high tower from a point A on the same level as the foot of the tower is 25^{\circ} . Find the angle of elevation of the top of the tower from a point B 20 \mathrm{~m} nearer to A from the foot of the tower.

1. Find the unknown angles and sides of the following triangles:
1. Find the unknown angles and sides of the following triangles:

1. Find the unknown angles and sides of the following triangles:

2. A man 18 \mathrm{dm} tall observes that the angle of elevation of the top of a tree at a distance of 12 \mathrm{~m} from him is 32 . What is the height of the tree?
2. A man  18 \mathrm{dm}  tall observes that the angle of elevation of the top of a tree at a distance of  12 \mathrm{~m}  from him is 32 . What is the height of the tree?

2. A man 18 \mathrm{dm} tall observes that the angle of elevation of the top of a tree at a distance of 12 \mathrm{~m} from him is 32 . What is the height of the tree?

Q.6 In a right triangle no angle is greater than(a) 90^{\circ} (b) 80^{\circ} (c) 60^{\circ} (d) 45^{\circ} Sahiwal Board 2013; Lahore Board 2008
Q.6 In a right triangle no angle is greater than(a)  90^{\circ} (b)  80^{\circ} (c)  60^{\circ} (d)  45^{\circ} Sahiwal Board 2013; Lahore Board 2008

Q.6 In a right triangle no angle is greater than(a) 90^{\circ} (b) 80^{\circ} (c) 60^{\circ} (d) 45^{\circ} Sahiwal Board 2013; Lahore Board 2008

Example 1: Solve the triangle A B C by using the law of cosine when a=7 b=3 c=5
Example 1: Solve the triangle  A B C  by using the law of cosine when  a=7 b=3 c=5

Example 1: Solve the triangle A B C by using the law of cosine when a=7 b=3 c=5

Q.13 \frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma} is law of(a) tangent(b) cosine(c) \sin e (d) cotangent
Q.13  \frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}  is law of(a) tangent(b) cosine(c)  \sin e (d) cotangent

Q.13 \frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma} is law of(a) tangent(b) cosine(c) \sin e (d) cotangent

Solve the triangle A B C if3. b=125 \gamma=53^{\circ} \alpha=47^{\circ}
Solve the triangle  A B C  if3.   b=125   \gamma=53^{\circ}   \alpha=47^{\circ}

Solve the triangle A B C if3. b=125 \gamma=53^{\circ} \alpha=47^{\circ}

1. Find the values of:\[\text { i) } \sin 53^{\circ} 40^{\prime}\]
1. Find the values of:\[\text { i) } \sin 53^{\circ} 40^{\prime}\]

1. Find the values of:\[\text { i) } \sin 53^{\circ} 40^{\prime}\]

Example 3: Prove that \frac{1}{r^{2}}+\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}=\frac{a^{2}+b^{2}+c^{2}}{\Delta^{2}}
Example 3: Prove that  \frac{1}{r^{2}}+\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}=\frac{a^{2}+b^{2}+c^{2}}{\Delta^{2}}

Example 3: Prove that \frac{1}{r^{2}}+\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}=\frac{a^{2}+b^{2}+c^{2}}{\Delta^{2}}

Solve the following triangles in which5. a=4584 b=5140 c=3624
Solve the following triangles in which5.  a=4584   b=5140   c=3624

Solve the following triangles in which5. a=4584 b=5140 c=3624

1. Find the unknown angles and sides of the following triangles:
1. Find the unknown angles and sides of the following triangles:

1. Find the unknown angles and sides of the following triangles:

7. Prove that in an equilateral trianglei) r: R: r_{1}=1: 2: 3
7. Prove that in an equilateral trianglei)  r: R: r_{1}=1: 2: 3
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7. Prove that in an equilateral trianglei) r: R: r_{1}=1: 2: 3

Q.2 Angle below the horizontal line is called :(a) Right angle(b) Oblique angle(c) Angle of depression(d) Angle of elevation
Q.2 Angle below the horizontal line is called :(a) Right angle(b) Oblique angle(c) Angle of depression(d) Angle of elevation

Q.2 Angle below the horizontal line is called :(a) Right angle(b) Oblique angle(c) Angle of depression(d) Angle of elevation

1. Find the values of:v) \cos 42^{\circ} 38^{\prime}
1. Find the values of:v)  \cos 42^{\circ} 38^{\prime}

1. Find the values of:v) \cos 42^{\circ} 38^{\prime}

9. Show that:ii) \frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}
9. Show that:ii)  \frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}

9. Show that:ii) \frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}

2. Find \theta if:iii) \cos \theta=0.5257
2. Find  \theta  if:iii)   \cos \theta=0.5257

2. Find \theta if:iii) \cos \theta=0.5257

4. The area of triangle is 2437 . If a=79 and c=97 then find angle \beta .
4. The area of triangle is 2437 . If  a=79  and  c=97  then find angle  \beta .

4. The area of triangle is 2437 . If a=79 and c=97 then find angle \beta .

1. Find the unknown angles and sides of the following triangles:
1. Find the unknown angles and sides of the following triangles:
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1. Find the unknown angles and sides of the following triangles:

Example 2: Solve the triangle A B C by half angle formula when\[a=283 b=317 c=428\]
Example 2: Solve the triangle  A B C  by half angle formula when\[a=283  b=317 c=428\]
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Example 2: Solve the triangle A B C by half angle formula when\[a=283 b=317 c=428\]

6. Find the smallest angle of the triangle A B C when a=37.34 b=3.24 c=35.06
6. Find the smallest angle of the triangle  A B C  when  a=37.34   b=3.24 c=35.06
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6. Find the smallest angle of the triangle A B C when a=37.34 b=3.24 c=35.06

Q.17 If the \triangle A B C is right angled then the law of cosines reduces to(a) law of sines(b) law of cosines(c) law of tangent(d) Pythagoras theorem
Q.17 If the  \triangle A B C  is right angled then the law of cosines reduces to(a) law of sines(b) law of cosines(c) law of tangent(d) Pythagoras theorem
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Q.17 If the \triangle A B C is right angled then the law of cosines reduces to(a) law of sines(b) law of cosines(c) law of tangent(d) Pythagoras theorem

5. Prove that:iii) r_{1}+r_{2}+r_{3}-r=4 R
5. Prove that:iii)   r_{1}+r_{2}+r_{3}-r=4 R
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5. Prove that:iii) r_{1}+r_{2}+r_{3}-r=4 R

Solve the right triangle A \dot{B} C in which \gamma=90^{\circ} 6. a=5429 c=6294
Solve the right triangle  A \dot{B} C  in which  \gamma=90^{\circ} 6.  a=5429  c=6294
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Solve the right triangle A \dot{B} C in which \gamma=90^{\circ} 6. a=5429 c=6294

4. A ladder leaning against a vertical wall makes an angle of 24^{\circ} with the wall. Its foot is 5 \mathrm{~m} from the wall. Find its length.
4. A ladder leaning against a vertical wall makes an angle of  24^{\circ}  with the wall. Its foot is  5 \mathrm{~m}  from the wall. Find its length.
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4. A ladder leaning against a vertical wall makes an angle of 24^{\circ} with the wall. Its foot is 5 \mathrm{~m} from the wall. Find its length.

12. Two forces of 40 N and 30 N are represented by \overrightarrow{A B} and \overrightarrow{B C} which are inclined at an angle of 147^{\circ} 25^{\prime \prime} . Find \overrightarrow{A C}^{-} the resultant of \overrightarrow{A B} and \overrightarrow{B C} .
12. Two forces of  40 N  and  30 N  are represented by  \overrightarrow{A B}  and  \overrightarrow{B C}  which are inclined at an angle of  147^{\circ} 25^{\prime \prime} . Find  \overrightarrow{A C}^{-}  the resultant of  \overrightarrow{A B}  and  \overrightarrow{B C} .
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12. Two forces of 40 N and 30 N are represented by \overrightarrow{A B} and \overrightarrow{B C} which are inclined at an angle of 147^{\circ} 25^{\prime \prime} . Find \overrightarrow{A C}^{-} the resultant of \overrightarrow{A B} and \overrightarrow{B C} .

Solve the right triangle A B C in which \gamma=90^{\circ} 7. \beta=50^{\circ} 10^{\prime} c=0.832
Solve the right triangle  A B C  in which  \gamma=90^{\circ} 7.  \beta=50^{\circ} 10^{\prime}  c=0.832
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Solve the right triangle A B C in which \gamma=90^{\circ} 7. \beta=50^{\circ} 10^{\prime} c=0.832

1. Find the values of:iv) \cot 33^{\circ} 50^{\prime}
1. Find the values of:iv)  \cot 33^{\circ} 50^{\prime}
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1. Find the values of:iv) \cot 33^{\circ} 50^{\prime}

Q.29 Circum radius of a circle is(a) R (b) r (c) r_{1} (d) r_{2}
Q.29 Circum radius of a circle is(a)  R (b)  r (c)  r_{1} (d)  r_{2}
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Q.29 Circum radius of a circle is(a) R (b) r (c) r_{1} (d) r_{2}

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