TRIANGLES

NO. 23

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TRIGONOMETRY A STUDY OF TRIANGLES

THE HISTORY OF TRIANGLES New Insights and breathtaking facts

HIPPARCHUS OR PYTHAGORAS Who's the father of Trigonometry?

Now Trending: Right Triangles

SOLVING RIGHT TRIANGLES Law of Sines and Laws of Cosines

www.bim.easyaccessmaterials.com ac@gmail.com Published monthly by AxC Inc. All Rights Reserved.

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CONTENT Summer Edition 2020 Issue 19

axc Vol. 1 Issue 19 may 2020 Introductory trigonometry Hipparchus of nicaea

. . . . . . . .

p. 4 p. 4

Pythagoras of samos Pythagorean theorem

. . . . . . . .

p. 6 p. 7

must know concepts: Special right triangles similar right triangles

. . . . . . . . . . . .

p. 7 p. 7

understanding the tangent trio studying sine and cosine ratios

. . . . . . . .

p. 8 p. 8

solving right triangles law of sine and laws of cosine

. . . . . . . .

p. 8 p. 9

word search extra practice Credits

. . . . . . . . . . . .

p. 11 p. 13 p. 15

EDITORS AMALIA FERNANDEZ VERA CASQUERO ST. MICHAELS SCHOOL, STO. DMGO. DOMINICAN REPUBLIC Phone: 1 800 564 6938

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INTRODUCTION TO TRIGONOMETRY THE STUDY OF TRIANGLES

Trigonometry is a branch of mathematics that studies

These studies were discovered by Egyptian, Babylonian,

relationships between side lengths and angles of triangle.

and Indian civilizations about 3000 years ago and have

The word trigonometry comes from the Greek words tri

been applied since then. Later on, the first trigonometric

(triangle) gon (sides) and metron (to measure). Its also

table was compiled by Hipparchus of Nicaea (180 – 125

concerned with specific functions of angles and their

BCE), who is now consequently known as "the father of

application to calculations.

trigonometry".

It is scientific study used to measure bearings and precise distances. Trigonometry uses the three basic functions of Sine, Cosine and Tangent to make its calculations. If two angles or two sides are given, they can be worked out to measure the rest of the details.

Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent

"ASTRONOMY IS THE FIRST OF WISDOM." - HIPPARCHUS

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There are six functions of an angle commonly used in

He was the first to tabulate the corresponding values of arc

trigonometry. Their names and abbreviations are sine

and chord for a series of angles. Although he was such a

(sin), cosine (cos), tangent (tan), cotangent (cot), secant

revolutionary mathematician, very little of his life was kept

(sec), and cosecant (csc). These six trigonometric functions

in record and most of his work relies on second-hand

in relation to a right triangle are displayed in the figure.

reports.

BALENCIAGA

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PYTHAGORAS OF SAMOS A GREEK MATHEMATICIAN Pythagoras one of the most influential Greek philosophers who made important developments in mathematics, astronomy, and the theory of music. He was born in Samos, Ionia, around 569 BC. Pythagoras is often described as the first pure mathematician. The theorem now known as Pythagorean Theorem was used by the Babylonians 1000 years earlier yet he is recognized for being the first to prove it. He is an extremely important figure in the development of mathematics yet we know very little about his achievements. Unlike many later Greek mathematicians, who have written many books still in existence, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

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The Pythagorean Theorem

special right triangles

The Pythagorean Theorem is one of the most

A special right triangle is a right triangle with some

famous theorems in mathematical history. It was

regular

first acknowledged in ancient Babylon and Egypt

the triangle easier, or for which simple formulas exist.

feature

that

makes

calculations

on

during the 1900s AC. However, it was not widely publicized

until

Pythagoras,

the

greek

mathematician, stated it explicitly.

The theorem describes the relationship between the side lengths of a right triangle. In the triangle, the length of the hypotenuse is equivalent to the length of its legs, making it c2=a2+b2.

On the other hand, the Converse Pythagorean Theorem states that if the longest side of a triangle is equal to the sum of the squares of the lengths of the other sides, it is a right triangle.

You can use your knowledge of the Pythagorean Theorem and the six trigonometric functions to solve a right triangle. Because a right triangle is a

For example, a right triangle may have angles that form

triangle with a 90 degree angle, solving a right

simple relationships, such as 45°–45°–90° or 30°–60°–90°.

triangle requires that you find the measures of one

These are called "angle-based" right triangles.

or both of the other angles.

The sides and angles of right triangles can be found

45°–45°–90° Triangle Theorem

using the trigonometric ratios. For each problem the

In this triangle, the hypotenuse is √2 times as long as

information given should be written down and

each of its legs. Hypotenuse = leg • √2.

substituted into one of the ratios, which can then be solved as an equation. Remember to use apply the

30°–60°–90° Triangle Theorem

Pythagorean Theorem if only the sides of the

In this triangle, the hypotenuse is twice as long as the

triangle are involved.

shorter leg, and the longer leg is √3 times as long as the

REAL LIFE APPLICATION

shorter leg. Hypotenuse = shorter leg • 2 and the longer leg = shorter leg √3.

Similar Right Triangles In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. In other words, the geometric mean of two positive numbers a and b is the positive number x that satisfies a/x = x/b.

Right Triangle Similarity Theorem If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original and to each other.

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Right Triangle Similarity Theorem

There is a rule that applies for the sine and cosine of

If the altitude is drawn to the hypotenuse of a right

complementary angles. The sine of an acute angle is

triangle, then the two triangles formed are similar to

equal to the cosine of its complement and vice versa. If

the original and to each other.

A and B are complementary angles then the following would be true:

Trigonometric Ratio A trigonometric ratio: is a ratio of the lengths of two

sin A = cos(90 degrees - A)= cos B

sides in a right triangle.

sin B= cos(90 degrees - B) = cos A cos A = sin(90 degrees - A) = sin B cos B= sin(90 degrees - B) = sin A Inverse Operation In order to find the measure of an angle in a right triangle, one can use the inverse trigonometric ratio.

For example let’s say that tan A = x then, tan-1 x is going to equal the measure of angle A. The same applies for Sine and Cosine. So, if sin A = x then, sin-1

The Tangent Ratio

x will equal measure of angle A. If cos A = x then, cos-

A Tangent Ratio is a trigonometric ratio for acute

1 x will equal the measure of angle A.

angles that involves the lengths of two legs of a right triangle. In order to find the tangent ratio of an angle you must do the following; put the length of the leg opposite to the angle over the length of the leg adjacent to the angle.

The Sine and Cosine Ratios The sine and cosine ratios are a trigonometric ratio for acute angles that involve the leg and hypotenuse of a right triangle. In oder to find the sine ratio of an angle one must put the length of the leg opposite to the angle over the length of the hypotenuse. On the other hand, to find the cosine ratio of an angle; you must put the length of the leg adjacent to the angle over the length of the hypotenuse.

C

Tan U = 4/3 Sin U = 4/5 Cos U = 3/5

Solving Right Triangles To solve a right triangle is to find all of it’s angles and it’s side measures. This can be done when you know two side lengths or one side length and the measure of

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one acute angle. Then, you use the trigonometric ratios

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and the inverse trigonometric ratios to find the missing measures.

B

8

3

U

The area of any given triangle can be found by half the product of the lengths of two of it’s sides multiplied by the sine of their included triangle. Trigonometric ratios and their inverse operations can only be used to solve right triangles. In order to solve any triangle one must know the following laws: Law of Sines and Law of Cosines.

Law of Sines

The Law of Cosines

The law of Sines can be used when two angles and the

The law of Cosines can be used when to sides and it’s

length of any sides are known or when the length of two

included triangle are known or when all three sides are

sides and the angle opposite to one of the two sides are

known. In a triangle ABC with side lengths a, b, c the

known. In a triangle ABC with side lengths a, b, c the

law applies as follows:

law would go as follows: a squared = b squared + c squared - 2bc cos A sin A over A = sin B over b = sin C over c

b squared = a squared + c squared - 2ac Cos B c squared = a squared + b squared - 2ab Cos C

Using the law of sines and the proportion. Fill in the values that you know. Use the given values, not those

The third side of a triangle when we know two sides

that you've determined yourself. Use a calculator to

and the angle between them (like the example

determine the values of the sines. Multiply each side by

above)the angles of a triangle when we know all three

the denominator under b to solve for that length.

sides (as in the following example)

Example:

Example:

C

A 114°

23 11

26

20

C

115°

B

A B

Steps

Steps

1. Sin B/ b= Sin A/ a

1. a squared= b squared + c squared - 2bc Cos A

2. Sin B/ 11= Sin 115°/ 20

2. a squared= 23 squared + 26 squared - 2(23)(26) cos 114°

3. Sin B= 11 Sin 115°/ 20

3. a squared= 1205 - 1196 cos 114°

4. Measure Angle B= 29.9°

4. a = 41.1 A

Important For both of the laws, the name of the sides and angles are the following:

b

c B

a

C

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REAL LIFE APPLICATION You can use the sine ratio in order to find the height of the building up to the ledge

Steps 1. tan 27째 = opposite/ adjacent 2. tan 27째 = x/ 45 3. 45 x tan 27째 = x 4. 22.9 = x Explanation 1. Write the ratio of sine 27째 2. Substitute the values 3. Multiply each side by 45 4. Use calculator to get your answer

The height of the building up to the ledge would be around 22.9 meters.

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WORD SEARCH

COSINE INVERSE COSINE LAW OF SINE RIGHT TRIANGLE

TRIGONOMETRIC RATIO GEOMETRIC MEAN INVERSE SINE PYTHAGOREAN THEOREM

SINE LAW OF COSINE PYTHAGOREAN TRIPLE TANGENT

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LE BOULANGERIE best cake in the best shape

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EXERCISES 1.

Find the Sine, Cosine and Tangent of angel c

2. Find the values of the missing variables using sine and cosine

3. Complete the following triangles

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NEW CAT-EYE Triangles on Trend

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CREDITS BIBLIOGRAPHIC REFERENCES

BestMaths Online. (n.d.). Retrieved June 02, 2020, from http://bestmaths.net/online/index.php/yearlevels/year-11/year-11-topics/trigonometricpythagoras/ Hughes, B., Ottman, L., Jordan, L., Landers, M., Hayes, A., Meery, B., & Fortgang, A. (2019, November 20). The Pythagorean Theorem and Trigonometry. Retrieved June 02, 2020, from https://www.ck12.org/trigonometry/the pythagorean-theorem-andtrigonometry/lesson/Pythagorean-Theoremfor-Solving-Right-Triangles-TRIG/ Intro to the Pythagorean theorem (video). (n.d.). Retrieved June 02, 2020, from https://www.khanacademy.org/math/basicgeo/basic-geometry-pythagorean-theorem/geopythagorean-theorem/v/the-pythagoreantheorem Huffman, C. (2018, October 17). Pythagoras.

Retrieved June 02, 2020, from https://plato.stanford.edu/entries/pythagoras / Jones, A. (2018, August 17). Hipparchus.

Retrieved June 02, 2020, from https://www.britannica.com/biography/Hippar chus-Greek-astronomer Free Easy Access Student Edition. (n.d.).

Retrieved June 02, 2020, from https://bim.easyaccessmaterials.com/index.php ?level=12.00 Math Is Fun. (2017, October 23). Law of Sines and Cosines. Retrieved June 02, 2020, from https://www.mathsisfun.com/algebra/trigcosine-law.html

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