JSS 2 SECOND TERM MATHEMATICS REVISION
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Angles and Polygons.
Types of angles and polygons
Angles are the measurements of the space between two lines, and polygons are shapes with multiple straight sides.
Some common types of angles include:
Right angle: A right angle measures 90 degrees, like the corner of a square.
Acute angle: An acute angle is less than 90 degrees, like the angle at the tip of a triangle.
Obtuse angle: An obtuse angle is greater than 90 degrees, like the angle at the bottom of a triangle.
Some common types of polygons include:
Triangle: A triangle has three sides and three angles, like a slice of pizza.
Square: A square has four sides and four right angles, like a piece of paper.
Pentagon: A pentagon has five sides and five angles, like a soccer ball.
Angles:
Straight angle: A straight angle measures 180 degrees, like a line drawn across a page.
Reflex angle: A reflex angle is greater than 180 degrees but less than 360 degrees, like the angle formed when you fold a piece of paper in half.
Complementary angles: Two angles are complementary if their sum equals 90 degrees, like the angles at the base of an isosceles triangle.
Polygons:
Hexagon: A hexagon has six sides and six angles, like a stop sign.
Octagon: An octagon has eight sides and eight angles, like a stop sign but with more sides.
Circle: Although not a polygon, a circle is a shape with infinite sides and no angles, like a wheel.
Additionally, polygons can be classified based on their number of sides, as follows:
Three sides: triangle
Four sides: quadrilateral
Five sides: pentagon
Six sides: hexagon
Seven sides: heptagon
Eight sides: octagon
Nine sides: nonagon
Ten sides: decagon
Evaluation
1. Which of the following is an example of an obtuse angle?
a. 45 degrees
b. 90 degrees
c. 120 degrees
d. 180 degrees
2. A triangle has how many sides?
a. 2
b. 3
c. 4
d. 5
3. Which of the following shapes has four right angles?
a. Triangle
b. Square
c. Pentagon
d. Hexagon
4. What is the sum of the interior angles of a pentagon?
a. 360 degrees
b. 540 degrees
c. 720 degrees
d. 900 degrees
5. Which of the following is an example of a reflex angle?
a. 45 degrees
b. 180 degrees
c. 225 degrees
d. 270 degrees
6. What is the name of a polygon with ten sides?
a. Hexagon
b. Decagon
c. Octagon
d. Nonagon
7. Which of the following is an example of a right angle?
a. 45 degrees
b. 90 degrees
c. 120 degrees
d. 180 degrees
8. What is the sum of the interior angles of a hexagon?
a. 360 degrees
b. 540 degrees
c. 720 degrees
d. 900 degrees
9. Which of the following shapes has five sides?
a. Triangle
b. Square
c. Pentagon
d. Hexagon
10. Which of the following is not a type of angle?
a. Acute
b. Obtuse
c. Square
d. Right
INEQUALITIES
Inequality is a comparison between two values, showing that one value is greater than, less than, or equal to the other value.
Here are some examples:
3 < 5 – This reads as “3 is less than 5.”
7 > 4 – This reads as “7 is greater than 4.”
2 ≤ 2 – This reads as “2 is less than or equal to 2.”
9 ≥ 9 – This reads as “9 is greater than or equal to 9.”
We use symbols to represent the relationship between the two values:
< (less than) – used to show that one value is smaller than the other.
(greater than) – used to show that one value is larger than the other.
≤ (less than or equal to) – used to show that one value is smaller than or equal to the other.
≥ (greater than or equal to) – used to show that one value is larger than or equal to the other.
We can also use variables in inequalities. Here’s an example:
x + 5 > 10 – This reads as “the sum of x and 5 is greater than 10.”
We can solve this inequality to find the value of x:
x > 5 – This reads as “x is greater than 5.”
Understanding inequalities is important in math and everyday life, as they help us make comparisons and decisions based on values.
Worked Samples
1. Solve the inequality 4x + 7 < 19.
Subtract 7 from both sides: 4x < 12
Divide both sides by 4: x < 3
The solution is x is less than 3.
2. Solve the inequality 3x – 5 > 10.
Add 5 to both sides: 3x > 15
Divide both sides by 3: x > 5
The solution is x is greater than 5.
3. Solve the inequality 2x + 1 ≤ 7.
Subtract 1 from both sides: 2x ≤ 6
Divide both sides by 2: x ≤ 3
The solution is x is less than or equal to 3.
4. Solve the inequality 2(x + 1) > 6.
Distribute the 2: 2x + 2 > 6
Subtract 2 from both sides: 2x > 4
Divide both sides by 2: x > 2
The solution is x is greater than 2.
5. Solve the inequality 3(x – 4) ≤ -6.
Distribute the 3: 3x – 12 ≤ -6
Add 12 to both sides: 3x ≤ 6
Divide both sides by 3: x ≤ 2
The solution is x is less than or equal to 2.
In each of these examples, we used the appropriate mathematical operations to isolate the variable and determine the range of values that satisfy the inequality.
Trigonometry
Trigonometry is the study of relationships between the sides and angles of triangles. There are three main trigonometric functions that are used to calculate these relationships: sine, cosine, and tangent.
Here are some examples:
Sine (sin) – the ratio of the length of the side opposite an angle to the length of the hypotenuse of a right-angled triangle.
sin(θ) = opposite/hypotenuse
For example, if the opposite side of an angle is 3 and the hypotenuse is 5, then sin(θ) = 3/5.
Cosine (cos) – the ratio of the length of the adjacent side to the length of the hypotenuse of a right-angled triangle.
cos(θ) = adjacent/hypotenuse
For example, if the adjacent side of an angle is 4 and the hypotenuse is 5, then cos(θ) = 4/5.
Tangent (tan) – the ratio of the length of the opposite side to the length of the adjacent side of a right-angled triangle.
tan(θ) = opposite/adjacent
For example, if the opposite side of an angle is 3 and the adjacent side is 4, then tan(θ) = 3/4.
Trigonometry can be used to solve problems involving angles and distances, such as finding the height of a building or the length of a bridge.
Understanding trigonometry is important in fields such as engineering, architecture, and physics.
Is trigonometry always used for right angle triangle?
Yes, trigonometry is always used for right-angled triangles. Trigonometric functions such as sine, cosine, and tangent are defined only for right-angled triangles. These functions relate the ratios of the sides of a right-angled triangle to the measures of its angles.
For example, in a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Trigonometry can be used to solve various problems related to right-angled triangles, such as finding missing angles or sides, or calculating distances and heights. However, if the triangle is not right-angled, trigonometry cannot be used directly. In such cases, other techniques, such as the law of sines or the law of cosines, may be used.
Evaluation
1. Which symbol is used to represent “less than or equal to”?
a. <
b. >
c. ≤
d. ≥
2. Solve the inequality 4x – 7 > 5. What is the solution?
a. x < 3
b. x > 3
c. x ≤ 3
d. x ≥ 3
3. What is the solution to the inequality 2x + 1 < 9?
a. x < 4
b. x > 4
c. x ≤ 4
d. x ≥ 4
4. Which symbol is used to represent “greater than or equal to”?
a. <
b. >
c. ≤
d. ≥
5. Solve the inequality 2x – 3 > 7. What is the solution?
a. x < 5
b. x > 5
c. x ≤ 5
d. x ≥ 5
6. What is the solution to the inequality 3x – 4 < 8?
a. x < 4
b. x > 4
c. x ≤ 4
d. x ≥ 4
7. Which symbol is used to represent “less than”?
a. <
b. >
c. ≤
d. ≥
8. Solve the inequality 5x + 3 ≥ 18. What is the solution?
a. x < 3
b. x > 3
c. x ≤ 3
d. x ≥ 3
9. What is the solution to the inequality 4x + 2 < 10?
a. x < 2
b. x > 2
c. x ≤ 2
d. x ≥ 2
10. Which symbol is used to represent “greater than”?
a. <
b. >
c. ≤
d. ≥
Worked Samples On Trigonometry
1. Find the value of sine of an angle in a right triangle where the opposite side is 3 and the hypotenuse is 5.
sin(θ) = opposite/hypotenuse
sin(θ) = 3/5
The value of sin(θ) is 0.6.
Find the value of cosine of an angle in a right triangle where the adjacent side is 4 and the hypotenuse is 5.
cos(θ) = adjacent/hypotenuse
cos(θ) = 4/5
The value of cos(θ) is 0.8.
2. Find the value of tangent of an angle in a right triangle where the opposite side is 3 and the adjacent side is 4.
tan(θ) = opposite/adjacent
tan(θ) = 3/4
The value of tan(θ) is 0.75.
3. Find the length of the hypotenuse in a right triangle where the opposite side is 4 and the adjacent side is 3.
Use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 4^2 + 3^2
hypotenuse^2 = 16 + 9
hypotenuse^2 = 25
hypotenuse = √25
The length of the hypotenuse is 5.
4. Find the measure of an angle in a right triangle where the opposite side is 3 and the adjacent side is 4.
Use the inverse tangent function: tan^-1(opposite/adjacent)
tan^-1(3/4) ≈ 36.87 degrees
The measure of the angle is approximately 36.87 degrees.
In each of these examples, we used the appropriate trigonometric function or formula to solve for a missing value or angle in a right triangle.
Evaluation
1. What is the value of sine of a 90-degree angle in a right triangle?
a. 0
b. 1
c. undefined
d. infinity
2. What is the value of cosine of a 0-degree angle in a right triangle?
a. 0
b. 1
c. undefined
d. infinity
3. What is the value of tangent of a 45-degree angle in a right triangle?
a. 0
b. 1
c. 0.5
d. undefined
4. In a right triangle, if the hypotenuse is 10 and the opposite side is 6, what is the value of sine of the angle opposite the opposite side?
a. 0.6
b. 0.8
c. 0.4
d. 0.2
5. In a right triangle, if the adjacent side is 4 and the hypotenuse is 5, what is the value of cosine of the angle adjacent to the adjacent side?
a. 0.6
b. 0.8
c. 0.4
d. 0.2
6. In a right triangle, if the opposite side is 3 and the adjacent side is 4, what is the value of tangent of the angle opposite the opposite side?
a. 0.75
b. 0.6
c. 1.25
d. 1.33
7. In a right triangle, if the hypotenuse is 5 and the adjacent side is 3, what is the value of sine of the angle adjacent to the adjacent side?
a. 0.6
b. 0.8
c. 0.4
d. 0.2
8. In a right triangle, if the opposite side is 3 and the hypotenuse is 5, what is the value of cosine of the angle opposite the opposite side?
a. 0.6
b. 0.8
c. 0.4
d. 0.2
9. In a right triangle, if the adjacent side is 4 and the hypotenuse is 5, what is the value of tangent of the angle opposite the adjacent side?
a. 0.8
b. 0.6
c. 1.25
d. 1.33
10. What is the measure of an angle in a right triangle if the opposite side is 3 and the adjacent side is 4?
a. 36.87 degrees
b. 45 degrees
c. 53.13 degrees
d. 60 degrees
Pythagoras rule
Hey there! Let’s learn about Pythagoras rule!
Pythagoras rule is a formula used to find the length of the sides of a right-angled triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Here’s an example:
In a right triangle with legs of length 3 and 4, what is the length of the hypotenuse?
The hypotenuse is the longest side of the triangle and is opposite the right angle.
Use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 3^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
hypotenuse = √25
The length of the hypotenuse is 5.
Understanding Pythagoras rule is important in solving problems related to right-angled triangles. It is also used in various fields, such as engineering and architecture.
Worked Samples On Pythagorean Theorem
1. Find the length of the hypotenuse in a right triangle where the legs have lengths 5 and 12.
Use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 5^2 + 12^2
hypotenuse^2 = 25 + 144
hypotenuse^2 = 169
hypotenuse = √169
The length of the hypotenuse is 13.
2. Find the length of one leg of a right triangle where the other leg has length 8 and the hypotenuse has length 10.
Use the Pythagorean theorem: leg^2 = hypotenuse^2 – opposite^2
leg^2 = 10^2 – 8^2
leg^2 = 100 – 64
leg^2 = 36
leg = √36
The length of the leg is 6.
3. Find the length of the hypotenuse in a right triangle where the legs have lengths 9 and 12.
Use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 9^2 + 12^2
hypotenuse^2 = 81 + 144
hypotenuse^2 = 225
hypotenuse = √225
The length of the hypotenuse is 15.
4. Find the length of one leg of a right triangle where the hypotenuse has length 17 and the other leg has length 8.
Use the Pythagorean theorem: leg^2 = hypotenuse^2 – opposite^2
leg^2 = 17^2 – 8^2
leg^2 = 289 – 64
leg^2 = 225
leg = √225
The length of the leg is 15.
5. Find the length of one leg of a right triangle where the hypotenuse has length 13 and the other leg has length 5.
Use the Pythagorean theorem: leg^2 = hypotenuse^2 – opposite^2
leg^2 = 13^2 – 5^2
leg^2 = 169 – 25
leg^2 = 144
leg = √144
The length of the leg is 12.
In each of these examples, we used the Pythagorean theorem to find a missing length of a right triangle, given the lengths of the other sides.
Evaluation
1. What is the Pythagorean theorem used for?
a. Finding the area of a circle
b. Finding the length of the hypotenuse in a right triangle
c. Finding the volume of a sphere
d. Finding the perimeter of a rectangle
2. What is the formula for the Pythagorean theorem?
a. a^2 + b^2 = c
b. a^2 + c^2 = b
c. b^2 + c^2 = a
d. a + b + c = 180
3. What is the length of the hypotenuse in a right triangle with legs of length 3 and 4?
a. 5
b. 6
c. 7
d. 8
4. What is the length of one leg of a right triangle where the other leg has length 6 and the hypotenuse has length 10?
a. 4
b. 6
c. 8
d. 10
5. What is the length of the hypotenuse in a right triangle with legs of length 5 and 12?
a. 13
b. 15
c. 17
d. 19
6. What is the length of one leg of a right triangle where the hypotenuse has length 20 and the other leg has length 16?
a. 12
b. 16
c. 18
d. 24
7. What is the length of the hypotenuse in a right triangle with legs of length 9 and 12?
a. 15
b. 18
c. 21
d. 24
8. What is the length of one leg of a right triangle where the hypotenuse has length 26 and the other leg has length 24?
a. 10
b. 12
c. 14
d. 16
9. What is the length of one leg of a right triangle where the hypotenuse has length 29 and the other leg has length 21?
a. 12
b. 20
c. 24
d. 28
10. What is the length of the hypotenuse in a right triangle with legs of length 7 and 24?
a. 25
b. 26
c. 29
d. 30
Geometry solid shapes like cylinder, cone, cuboid, sphere, prism etc
In geometry, there are many solid shapes such as cylinder, cone, cuboid, sphere, and prism. These shapes have different properties and are used in many fields, such as architecture and engineering.
Here are some examples of these shapes:
Cylinder: A cylinder is a three-dimensional shape with two circular bases that are parallel to each other. It looks like a can of soda or a tube. Examples of cylinders include cans, pipes, and pencils.
Cone: A cone is a three-dimensional shape with a circular base that tapers to a point called the apex. It looks like an ice cream cone or a party hat. Examples of cones include traffic cones and volcano cones.
Cuboid: A cuboid is a three-dimensional shape with six rectangular faces. It looks like a box or a rectangular prism. Examples of cuboids include shoeboxes and dice.
Sphere: A sphere is a three-dimensional shape with a curved surface that is equidistant from a fixed point called the center. It looks like a ball or a globe. Examples of spheres include tennis balls and planets.
Prism: A prism is a three-dimensional shape with two parallel and congruent bases that are connected by rectangular faces. It looks like a box with a sloping top. Examples of prisms include triangular prisms and hexagonal prisms.
Understanding the properties of these solid shapes is important in solving problems related to their volume, surface area, and other characteristics. It is also useful in many everyday situations, such as calculating the amount of water that can be held in a cylindrical tank or finding the volume of a box.
Worked Samples
1. Find the surface area of a cylinder with a radius of 4 cm and a height of 8 cm.
The surface area of a cylinder is given by: 2πr² + 2πrh, where r is the radius and h is the height.
Surface area = 2π(4)² + 2π(4)(8)
Surface area = 2π(16) + 2π(32)
Surface area = 32π + 64π
Surface area = 96π square cm
The surface area of the cylinder is approximately 301.59 square cm.
2. Find the volume of a cone with a radius of 6 cm and a height of 10 cm.
The volume of a cone is given by: (1/3)πr²h, where r is the radius and h is the height.
Volume = (1/3)π(6)²(10)
Volume = (1/3)π(36)(10)
Volume = 120π cubic cm
The volume of the cone is approximately 376.99 cubic cm.
3. Find the surface area of a cube with an edge length of 5 cm.
The surface area of a cube is given by: 6a², where a is the edge length.
Surface area = 6(5)²
Surface area = 6(25)
Surface area = 150 square cm
The surface area of the cube is 150 square cm.
4. Find the volume of a sphere with a radius of 3 cm.
The volume of a sphere is given by: (4/3)πr³, where r is the radius.
Volume = (4/3)π(3)³
Volume = (4/3)π(27)
Volume = 36π cubic cm
The volume of the sphere is approximately 113.1 cubic cm.
5. Find the total surface area of a rectangular prism with length 8 cm, width 5 cm, and height 6 cm.
The total surface area of a rectangular prism is given by: 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
Surface area = 2(8)(5) + 2(8)(6) + 2(5)(6)
Surface area = 80 + 96 + 60
Surface area = 236 square cm
The total surface area of the rectangular prism is 236 square cm.
In each of these examples, we used the formulas for finding the volume and surface area of various solid shapes, such as cylinders, cones, cubes, spheres, and rectangular prisms.
Evaluation
1. What is the formula for finding the surface area of a cylinder?
a. πr²
b. 2πrh
c. 2πr² + 2πrh
d. (1/3)πr²h
2. What is the formula for finding the volume of a cone?
a. πr²
b. (1/3)πr²h
c. 2πr² + 2πrh
d. (1/3)πr³h
3. What is the formula for finding the surface area of a cube?
a. 6a
b. 6a²
c. 4πr²
d. πr³
4. What is the formula for finding the volume of a sphere?
a. (4/3)πr³
b. 2πr²
c. (1/3)πr²h
d. 6a²
5. What is the formula for finding the surface area of a rectangular prism?
a. 2lw + 2lh + 2wh
b. 4πr²
c. 2πrh
d. (1/3)πr²h
6. Which of the following shapes has a circular base?
a. Cone
b. Cube
c. Rectangular prism
d. Pyramid
7. Which of the following shapes has a curved surface?
a. Cube
b. Sphere
c. Rectangular prism
d. Pyramid
8. Which of the following shapes has two parallel and congruent circular bases?
a. Cone
b. Cube
c. Rectangular prism
d. Pyramid
9. What is the formula for finding the volume of a cylinder?
a. πr²
b. 2πrh
c. 2πr² + 2πrh
d. πr²h
10. Which of the following shapes has the same length, width, and height?
a. Cone
b. Cube
c. Rectangular prism
d. Pyramid
Angles and Polygons.
