Trigonometry Concept

What is Trigonometry?

Trigonometry comes from Greek, "trigon" meaning triangle and "metron" meaning measure. So, trigonometry is simply the study of the relationship between sides and angles in triangles.

Recalling the Pythagorean Theorem

Before we dive into trigonometry concepts, we need to recall the Pythagorean theorem that applies to right triangles:

Where:

• $a$ and $b$ are the lengths of the two sides that form the right angle
• $c$ is the length of the hypotenuse

This theorem becomes an important foundation in trigonometry because many trigonometric calculations involve right triangles.

Understanding Ratios and Proportions

The concept of ratio or proportion is very important in trigonometry. A ratio is the relationship between two values expressed as a proportion.

For example, imagine you are observing a tree and a ruler under sunlight. If the shadow of the ruler is 3 cm and the height of the ruler is 6 cm, while the shadow of the tree is 150 cm, we can find the height of the tree using ratio:

So the tree height = $6 \div 3 \times 150 = 300 \text{ cm} = 3 \text{ meters}$

This is the power of ratios in everyday life!

Similar Triangles

Two triangles are said to be similar if they meet one of these conditions:

• The three angles in the triangles are equal, or
• The three sides of the triangles are proportional (have the same ratio)

Similar triangles have the same shape even though their sizes may differ. It's like a photo that's enlarged or reduced - the proportions remain the same!

Applications of Similar Triangles

The concept of similar triangles is widely used in everyday life:

1. Height measurement: By utilizing shadows, we can measure the height of buildings or trees without having to climb them.

2. Mapping: Cartographers (map makers) use the concept of similar triangles to create accurate maps with specific scales.

3. Architecture: Architects use this concept to design building models that are similar to the actual buildings.

Trigonometric Ratios

Trigonometric ratios are the ratios between sides in a right triangle related to a specific angle. This is the heart of trigonometry!

Imagine a right triangle with an angle θ (theta). We name its sides as:

• Hypotenuse: the side opposite to the right angle
• Opposite side: the side opposite to angle θ
• Adjacent side: the side adjacent to angle θ (other than the hypotenuse)

From these three sides, we get six basic trigonometric ratios:

1. Sine (sin):

   $$\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$$

2. Cosine (cos):

   $$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$$

3. Tangent (tan):

   $$\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$$

4. Cosecant (csc):

   $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite side}}$$

5. Secant (sec):

   $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent side}}$$

6. Cotangent (cot):

   $$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent side}}{\text{opposite side}}$$

Why Are Trigonometric Ratios Always the Same?

The interesting thing about trigonometric ratios is that their values are always the same for the same angle, regardless of the size of the right triangle. This is because triangles with the same angles are similar!

Let's understand this concept more deeply:

Imagine we have two right triangles with the same angle (for example $\theta = 30^\circ$) but different sizes. Because both triangles have the same angle, they are similar. In similar triangles, the ratios of their sides are always proportional.

Simple example:

• Triangle A has an opposite side of 3 cm and a hypotenuse of 6 cm
• Triangle B has an opposite side of 5 cm and a hypotenuse of 10 cm

If we calculate $\sin \theta$ for both triangles:

• $\sin \theta$ (triangle A) = $\frac{3}{6} = 0.5$
• $\sin \theta$ (triangle B) = $\frac{5}{10} = 0.5$

Even though the triangle sizes are different, the value of sin θ remains the same! This is why we can create trigonometric value tables that are universally applicable.

The unit circle visualization above shows how trigonometric ratio values are determined by the angle, not by the size of the triangle. On the unit circle (with radius 1), the coordinates of a point on the circle directly represent the sin and cos values for that angle.

This concept is very important because it allows us to use the same trigonometric values to solve problems of different sizes, as long as the angle is the same.

Applications of Trigonometry in Real Life

Trigonometry may seem complex, but it's actually used in many aspects of everyday life:

Measuring Height Using Shadows

Have you ever noticed shadows? The shadow of a tall person will be longer than the shadow of a short person at the same time. However, the ratio of height to shadow length remains the same as long as the position of the sun is the same!

This principle was used by ancient scientists to make extraordinary measurements. For example, a Greek mathematician named Eratosthenes successfully measured the circumference of the earth quite accurately about 2000 years ago, just by observing shadow differences in different locations.

Navigation

Ancient sailors used trigonometry for navigation by measuring the angle between the horizon and stars.

Architecture and Construction

Architects use trigonometry to calculate roof slopes, tower heights, and many structural aspects of buildings.

Electronics and Waves

Sound waves, radio waves, and alternating electric currents can be modeled using sine and cosine functions.

Simple Experiment

You can try a simple experiment to better understand trigonometry:

1. Measure your height and the length of your shadow under sunlight (make sure the time is fixed, for example 10 AM)

2. Calculate the ratio of your height to your shadow length

3. Use this ratio to estimate the height of other objects (trees, poles, buildings) by measuring just their shadow length

4. Mark your position, the object's position, and draw a line from the top of your head to the end of your shadow

5. Measure the angle formed with a protractor. This angle will be the same for all objects at the same time!

With this simple experiment, you have applied the basic principles of trigonometry!