The document explains the Pythagorean theorem and provides examples of its use. It defines key terms like right triangle, hypotenuse, and how angles and sides are labeled. The theorem states that for a right triangle with sides a, b, c, where c is the hypotenuse, a^2 + b^2 = c^2. Examples show setting up and solving equations using the theorem to find missing side lengths of right triangles. The final example calculates the length of cloth needed to make a tent with a 4m high, 3m wide opening.
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
Define the sine, cosine, and tangent ratios and their inverses
Find the measure of a side given a side and an angle
Find the measure of an angle given two sides
Use trig ratios to solve problems
the slides uploaded above are a way to learn and master the basics of trigonometry, the explain and define all three triangles and the hypotenuse rule, which is the Pythagoras.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
the slides uploaded above are a way to learn and master the basics of trigonometry, the explain and define all three triangles and the hypotenuse rule, which is the Pythagoras.
Trigonometry is a field of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Though many modern students who pursue this career are unaware of its history, it has always been important to engineers. Pythagoras, the Pythagorean Theorem and Archimedes’ The Great Bridge are just a few of the timeless concepts that the mathematics of triangles can be applied to. Trigonometry has two primary components: arithmetic and geometry. Geometry describes geometric relationships between lines and angles. Arithmetic is the study of multiplication, division, integration, and multiplication.
The data is present below the pictures so as to edit it as per your needs. I wanted to use good fonts and this was the only way i could do it as the fonts would not be available on your computer.
The Law of Sines is a principle of trigonometry stating that the length of the sides of any triangle are proportional to the sines of the opposite angles.
These slides contain the pathagorean theorem and right trinagles. How to prove the oathagorean theorem and how to vind the area of triangles by the pathagorean theorem. There are some slides that explains that how the pathagorean theorem was discovrers. Some slides explain the pathagorean triple theorem and c^2=a^2 + b^2.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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MWA 10 7.1 Pythagorean
1. The Pythagorean Theorem
Slide 1
Terminology:
• Right Triangle – a triangle that has a right angle
• Hypotenuse – the side of the triangle that is directly across from the right angle.
2. Slide 2
Labelling Parts of a Right Triangle
Angles
• are labelled with uppercase letters. ie: A, B, C
• the letter C is most commonly used to label the right angle
3. Slide 3
Labelling Parts of a Right Triangle
Angles
• are labelled with uppercase letters. ie: A, B, C
• the letter C is most commonly used to label the right angle
Sides
• are labelled with lowercase letters. ie: a, b, c
• the side has the same letter as the angle across from it
• the letter c is most commonly used to label the hypotenuse (which is
directly across from the right angle)
4. Slide 4
Pythagorean Theorem
The Pythagorean theorem states the relationship among the sides
of a right triangle. Given a right triangle ABC with right angle C, the
Pythagorean theorem states the following:
2 2 2
a b c
5. Slide 5
Pythagorean Theorem
The Pythagorean theorem states the relationship among the sides
of a right triangle. Given a right triangle, the Pythagorean theorem
can also be shown as:
2 2 2
1 2leg leg hypotenuse
leg1
leg2
6. Example 1:
Use the Pythagorean theorem to find the length of the missing
side of the triangle to the nearest tenth of a unit.
Slide 6
7. Example 1:
Use the Pythagorean theorem to find the length of the missing
side of the triangle to the nearest tenth of a unit.
Slide 7
Identify the hypotenuse.
Set up the formula and solve.
2 2 2
r p q
2 2 2
3.8 5.2 q
2 2 2
3.8 5.2 q
2
14.44 27.04 q
2
41.48 q
2
41.48 q
6.44q m
Note: When you take a square root of a number,
normally you would include since you do not know if
the answer will be positive or negative. When dealing
with distance, the answer will always be positive.
8. Slide 8
Example 2:
Use the Pythagorean theorem to find the lengths of the missing
sides of the triangles to the nearest tenth of a unit.
9. Identify the hypotenuse.
Set up the formula and solve.
Slide 9
Example 2:
Use the Pythagorean theorem to find the lengths of the missing
sides of the triangles to the nearest tenth of a unit.
2 2 2
z y x
2 22
6.9 12.8z
2
47.61 163.84z
2
116.23z
2
116.23z
10.78z Reminder – lengths are always
positive so the sign is not required.
10. Slide 10
Example 3:
In the Old West, settlers often fashioned tents out of a piece of
cloth thrown over tent poles and then secured to the ground with
stakes forming an isosceles triangle. How long would the cloth
have to be so that the opening of the tent was 4 meters high
and 3 meters wide?
11. Slide 11
Example 3:
In the Old West, settlers often fashioned tents out of a piece of
cloth thrown over tent poles and then secured to the ground with
stakes forming an isosceles triangle. How long would the cloth
have to be so that the opening of the tent was 4 meters high
and 3 meters wide?
Draw a diagram.
3
4
x
Set up the equation.
2 2 2
leg leg hyp
2 2 2
4 3 x
2
16 9 x
2
25 x
2
25 x
5 x
Calculate length of cloth.
3 + 3
55
5 + 5 + 3 + 3 = 16 m
The cloth should be 16 m long.