📐 Level 1 - Topic 4: Basic Trigonometric Ratios (Sine, Cosine, Tangent) 📊

1) What are Ratios? - A Quick Review 📈

Before we dive into trigonometric ratios, let's quickly remind ourselves what a ratio is in mathematics. A ratio is simply a way to compare two quantities by division. It tells us how much of one thing there is compared to another.

A Ratio is a comparison of two quantities using division. It can be expressed as a fraction, using a colon, or in words (e.g., "ratio of a to b").

Examples of Ratios in Everyday Life:

Speed: Speed is a ratio of distance traveled to time taken (e.g., miles per hour, kilometers per second). It compares distance and time.
Scale on a Map: A map scale (like 1:100,000) is a ratio comparing a distance on the map to the corresponding distance on the ground.
Gear Ratios in Mechanics: Gear ratios in machines compare the number of rotations of one gear to another.
Recipes: Recipes often use ratios to describe proportions of ingredients (e.g., "2 parts flour to 1 part sugar").

Ratios are fundamental in mathematics and science to express relationships between different quantities. Now, let's see how ratios are used in trigonometry.

2) Introducing Trigonometric Ratios: Sine, Cosine, Tangent - Functions of Angles 📐

In trigonometry, we are interested in relationships between angles and ratios of lengths. Trigonometric ratios are specific ratios that relate angles to the sides of geometric figures. The three fundamental trigonometric ratios are: sine, cosine, and tangent.

Trigonometric Ratios (Sine, Cosine, Tangent) are functions of angles that represent ratios of certain lengths related to those angles. They are fundamental building blocks of trigonometry.

Sine, Cosine, Tangent as Functions:

Think of sine, cosine, and tangent as functions. You input an angle, and the function outputs a ratio (a number). For example:

Sine Function (sin): Takes an angle \( \theta \) as input and gives a ratio \( \sin(\theta) \) as output.
Cosine Function (cos): Takes an angle \( \theta \) and gives a ratio \( \cos(\theta) \).
Tangent Function (tan): Takes an angle \( \theta \) and gives a ratio \( \tan(\theta) \).

What do these ratios represent?

Initially, it might seem abstract. But these ratios are not arbitrary. They capture fundamental relationships that are consistent for a given angle, regardless of the size of the geometric figure. We've already seen in Topic 3 with the unit circle how sine and cosine relate to the coordinates on the circle. In the next topic, we'll connect these ratios specifically to the sides of right-angled triangles, making them more concrete and applicable to calculations.

Key Idea: Sine, cosine, and tangent are *functions* of angles. For every angle, there's a specific sine value, a specific cosine value, and a specific tangent value (except for angles where tangent is undefined). These values are *ratios*.

3) Initial Look at Sine and Cosine - Using the Unit Circle (Recap) 🌐

Let's briefly revisit the unit circle from Topic 3 to get an initial sense of what these ratios are related to. Recall that for any angle \( \theta \) in standard position, the point where the terminal side intersects the unit circle is \( (\cos(\theta), \sin(\theta)) \).

Sine and Cosine in the Unit Circle Context:

Cosine \( (\cos(\theta)) \) as x-coordinate: The cosine of angle \( \theta \) is the x-coordinate of the point on the unit circle. It represents the horizontal position.
Sine \( (\sin(\theta)) \) as y-coordinate: The sine of angle \( \theta \) is the y-coordinate of the point on the unit circle. It represents the vertical position.

In this unit circle context, \( \cos(\theta) \) and \( \sin(\theta) \) are indeed *ratios*. Although it's not immediately obvious as a ratio of lengths in a triangle *yet*, think of the unit circle. The x and y coordinates are scaled relative to the radius of 1. [*(Think about how x and y coordinates in unit circle can be interpreted as ratios... perhaps expand on this slightly if needed for Topic 4 level)*]

4) Tangent - The Slope Connection ⛰️

What about the tangent function, \( \tan(\theta) \)? In the unit circle context, \( \tan(\theta) \) has a very important geometric interpretation: it's related to the slope of the terminal side of the angle.

Tangent and Slope:

For an angle \( \theta \) in standard position and a point \( (x, y) = (\cos(\theta), \sin(\theta)) \) on the unit circle (where \( x \neq 0 \)), the tangent of \( \theta \) is the slope of the line passing through the origin and the point \( (x, y) \).

\( \tan(\theta) = \text{Slope of Terminal Side} = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)} \) (for \( \cos(\theta) \neq 0 \))

Recall that slope is "rise over run," or \( \frac{\text{vertical change}}{\text{horizontal change}} \). In the unit circle, the vertical change is the y-coordinate (sine), and the horizontal change is the x-coordinate (cosine). Thus, tangent is the ratio of sine to cosine, and geometrically, it's the slope.

Undefined Tangent: Notice that tangent is undefined when \( \cos(\theta) = 0 \), because division by zero is undefined. This happens when the terminal side is vertical (angles like \( 90^\circ \) or \( 270^\circ \) or \( \frac{\pi}{2}, \frac{3\pi}{2} \) radians), where the slope is undefined.

5) Why are Trigonometric Ratios Important? - A Preview 🚀

You might be wondering, "Why are these trigonometric ratios so important?" Here's a glimpse into their significance:

Relate Angles and Lengths: Trigonometric ratios provide a fundamental link between angles and lengths. They allow us to use angles to calculate distances and heights, and vice versa.
Solving Triangles: They are essential for "solving triangles"—finding unknown angles and side lengths in triangles, especially right triangles (as we'll see in the next topic).
Modeling Periodic Phenomena: Sine and cosine functions are used to model periodic phenomena in nature and engineering: oscillations, waves, sound, light, alternating current, and many other cyclical processes.
Navigation, Physics, Engineering: Trigonometry is applied extensively in navigation (GPS, surveying), physics (mechanics, optics, waves), engineering (civil, mechanical, electrical), computer graphics, and many other fields.

In the upcoming topics, you'll see these applications unfold as you learn more about trigonometry!

6) Practice Questions 🎯

6.1 Fundamental – Basic Understanding

1. What is a ratio in mathematics? Give a real-world example of a ratio.

2. Define "trigonometric ratios" in general terms. What kind of quantities do they relate?

3. What are the three basic trigonometric ratios we are introducing in this topic?

4. Think of sine, cosine, and tangent as functions. What is the input to these functions? What is the output?

5. On the unit circle, what does the value of \( \cos(\theta) \) represent geometrically? What about \( \sin(\theta) \)?

6. How is \( \tan(\theta) \) related to \( \sin(\theta) \) and \( \cos(\theta) \)? Write the formula.

7. Explain the relationship between \( \tan(\theta) \) and the slope of the terminal side of the angle \( \theta \) in standard position.

8. For what angles is \( \tan(\theta) \) undefined? Why?

9. List at least three areas where trigonometric ratios are used in real-world applications.

10. True or False: Trigonometric ratios are just arbitrary numbers associated with angles, they don't represent any real relationships. Explain your answer.

6.2 Challenging – Conceptual & Exploratory 💪🚀

1. We've seen sine and cosine in the context of the unit circle and tangent related to slope. Can you think of other geometric interpretations of sine, cosine, or tangent beyond what's been directly described so far? (Think broadly about triangles, lines, distances).

2. Why do you think these *specific* ratios (sine, cosine, tangent) became so fundamental in mathematics and science, as opposed to other possible ratios you could define related to angles? Is there something particularly "special" about them?

3. Consider an angle \( \theta \) in the first quadrant \( (0^\circ < \theta < 90^\circ) \). As \( \theta \) increases, how do you think the values of \( \sin(\theta) \), \( \cos(\theta) \), and \( \tan(\theta) \) might change? (Think qualitatively - increase, decrease, stay the same, etc., based on their geometric interpretations).

4. We defined tangent as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Are there any angles for which \( \sin(\theta) = \cos(\theta) \)? If so, what would be the value of \( \tan(\theta) \) for such angles?

5. (Open-ended) If you were to invent a *new* trigonometric ratio, what kind of relationship or geometric property related to angles might you want it to capture? Describe what your new ratio would represent in your own terms.

7) Summary 🎉

Ratios: Comparison of two quantities using division. Common in math and everyday life.
Trigonometric Ratios (Sine, Cosine, Tangent): Functions of angles. Fundamental ratios in trigonometry, relating angles to lengths.
Sine and Cosine (Unit Circle): \( \cos(\theta) \) is x-coordinate, \( \sin(\theta) \) is y-coordinate of point on unit circle for angle \( \theta \).
Tangent (Slope): \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Geometrically, tangent is the slope of the terminal side of angle \( \theta \) in standard position.
Importance Preview: Trigonometric ratios link angles and lengths, used in solving triangles, modeling periodic phenomena, and numerous applications in science and engineering.

Excellent! You've now started your journey into understanding trigonometric ratios! You've been introduced to sine, cosine, and tangent as fundamental functions of angles and have explored their initial geometric interpretations. In the next topic, we'll see how these ratios become incredibly powerful when applied to right triangles! 📊📐🚀