Key Concepts

Trigonometry II: Unlocking the Secrets of Angles and Circles!

Habari mwanafunzi! Welcome to Trigonometry II. Remember in Form 2 when we learned how to find the height of a flagpole using its shadow? That was just the beginning! We used SOH CAH TOA, which is fantastic for right-angled triangles. But what about the real world? Angles are everywhere, and they are not always less than 90 degrees.

Imagine you are an engineer working on the new Nairobi Expressway. You need to calculate the precise curve of an exit ramp. Or think of a Safaricom engineer aligning a dish on a mast to connect to another one miles away. These calculations need a deeper understanding of angles – angles of all sizes! That's what we are going to learn today. Let's build on what you know and become masters of the circle!

1. A Quick Refresher: Our Old Friend SOH CAH TOA

Before we explore new lands, let's remember the tools that got us here. For any right-angled triangle, we have our three basic trigonometric ratios. Let's look at a triangle with angle θ (theta):

        /|
       / |
      /  |
   H /   | O (Opposite)
    /    |
   /_____|
  θ
    A
(Adjacent)

Our magic phrase is, and always will be, SOH CAH TOA:

SOH ->  Sine(θ) = Opposite / Hypotenuse

CAH ->  Cosine(θ) = Adjacent / Hypotenuse

TOA ->  Tangent(θ) = Opposite / Adjacent

This is our foundation. It's solid, reliable, and we will now see how it applies to a much bigger picture.

2. The Mighty Unit Circle: Our New Super-Tool!

Okay, get ready for the most important concept in Trig II: the Unit Circle. What is it? It’s simply a circle drawn on a Cartesian (x-y) plane with its center at the origin (0,0) and a radius of exactly 1 unit.

Why is this so powerful? Because it connects angles directly to (x, y) coordinates!

            (0,1)
              |
              |
      (-1,0)-----|-----(1,0)
              |
              |
            (0,-1)

Now, let's place a triangle inside it. If we draw a radius from the center (0,0) to any point (x,y) on the circle, it creates an angle θ with the positive x-axis. Look what happens:

          y-axis
           |
           |    /
           |   /
           |  /  r=1
         y | /
           |/_____θ_
           O   x   x-axis
        (x,y)

Using SOH CAH TOA on this new triangle:

• The Hypotenuse is the radius, which is 1.
• The side Adjacent to θ is the x-coordinate.
• The side Opposite to θ is the y-coordinate.

Now let's apply our formulas:

cos(θ) = Adjacent / Hypotenuse = x / 1  =>  x = cos(θ)

sin(θ) = Opposite / Hypotenuse = y / 1  =>  y = sin(θ)

This is a HUGE idea! For any point (x, y) on the unit circle, the x-coordinate is the cosine of the angle, and the y-coordinate is the sine of the angle. This allows us to find sin and cos for angles bigger than 90°, like 150°, 240°, or even 330°!

Image Suggestion: A vibrant and colorful educational diagram of the Unit Circle on a Cartesian plane. The circle should have a radius labeled 'r=1'. A point (x,y) is marked in the first quadrant, with a right-angled triangle formed by dropping a line to the x-axis. The sides should be labeled 'x', 'y', and '1'. The angle θ should be clearly marked from the positive x-axis. The labels 'x = cos(θ)' and 'y = sin(θ)' should be prominent.

3. Exploring the Four 'Maeneo' (The Quadrants)

The x and y axes divide our circle into four sections, or quadrants. Just like a shamba can be divided into four plots, our circle has four 'maeneo'. The sign (+ or -) of sin, cos, and tan changes depending on which quadrant the angle falls in.

           |
    II     |      I
  (180°-90°)|  (0°-90°)
           |
  ---------O---------> x-axis
           |
    III    |      IV
(180°-270°) | (270°-360°)
           |

How do we remember which ratio is positive where? We use a simple phrase: "All Students Take Chemistry" or ASTC.

• Quadrant I (0°-90°): All ratios (Sin, Cos, Tan) are positive.
• Quadrant II (90°-180°): Only Sine is positive. (Cos and Tan are negative).
• Quadrant III (180°-270°): Only Tangent is positive. (Sin and Cos are negative).
• Quadrant IV (270°-360°): Only Cosine is positive. (Sin and Tan are negative).

Example: What is the sign of cos(130°)?

130° is between 90° and 180°, so it's in Quadrant II. According to ASTC, only Sine is positive in Quadrant II. Therefore, cos(130°) must be negative. Try it on your calculator!

4. The VIPs: Special Angles (30°, 45°, 60°)

Some angles are special because we can find their exact trigonometric ratios without a calculator! These are our VIPs: 30°, 45°, and 60°. We derive them from two simple geometric shapes.

For 45°: We use an isosceles right-angled triangle with sides of length 1.

      /|
   √2/ | 1
    /  |
   /___|
  45°
    1

For 30° and 60°: We take an equilateral triangle with side length 2 and cut it in half.

      /|
     / |
    2/  | √3
    /   |
   /____|
  60°
    1

From these triangles, we get the following exact values. You should memorize these!

+--------+-------------+-------------+-------------+
| Angle  |    sin(θ)   |    cos(θ)   |    tan(θ)   |
+--------+-------------+-------------+-------------+
|  30°   |     1/2     |    √3 / 2   |    1 / √3   |
|  45°   |    1 / √2   |    1 / √2   |      1      |
|  60°   |    √3 / 2   |     1/2     |     √3      |
+--------+-------------+-------------+-------------+

5. Radians: A New Currency for Angles

So far, we've measured angles in degrees. But in advanced mathematics and science, we often use another unit called radians. Think of it like using Kenyan Shillings versus US Dollars – they measure the same thing (value) but use a different scale.

The key conversion you need to know is:

180° = π radians

With this, we can convert anything!

• To convert degrees to radians: Multiply by (π / 180)
• To convert radians to degrees: Multiply by (180 / π)

Calculation Example: Convert 60° to radians.

Step 1: Start with the angle in degrees.
   60°

Step 2: Multiply by the conversion factor (π / 180).
   60 * (π / 180)

Step 3: Simplify the fraction.
   = 60π / 180
   = 1π / 3
   = π/3 radians

So, 60° is the same as π/3 radians.

6. The Unbreakable Rule: The Pythagorean Identity

Finally, there is one formula in trigonometry that is so important, it’s like a law. It's called the Pythagorean Identity and it comes directly from the Pythagorean theorem applied to our unit circle.

Remember on the unit circle: x = cos(θ) and y = sin(θ). And the Pythagorean theorem for that triangle is x² + y² = 1².

If we substitute our trig values into the theorem, we get:

(cos(θ))² + (sin(θ))² = 1

Which we write as:

sin²θ + cos²θ = 1

This identity is always true, for any angle θ! It's incredibly useful because if you know the sine of an angle, you can find its cosine without even knowing the angle itself!

Phew! That was a lot, but you did great. We have journeyed from the simple right-angled triangle to the powerful unit circle. We've explored the four quadrants, met the VIP angles, learned a new language called radians, and discovered the unbreakable Pythagorean Identity. These are the key concepts that will unlock the rest of trigonometry for you.

Keep practicing, draw the unit circle until you can see it in your sleep, and don't be afraid to ask questions. Kazi nzuri!