Welcome to the World of Trigonometry!

Hello! Today, we are diving into Trigonometry. You might remember triangles from lower secondary, but in Additional Mathematics, we take it a step further. We explore how angles and side ratios behave like waves and how we can use "identities" (mathematical rules) to solve complex puzzles. Think of this chapter as building a "toolkit" for navigation, engineering, and even music production!

Don't worry if this seems tricky at first—we will break it down piece by piece. Let's get started!

1. The Six Trigonometric Functions

You already know Sine (sin), Cosine (cos), and Tangent (tan). Now, meet their "reciprocal" partners. These are just the flipped versions of the original three!

  • Cosecant (cosec A) = \( \frac{1}{\sin A} \)
  • Secant (sec A) = \( \frac{1}{\cos A} \)
  • Cotangent (cot A) = \( \frac{1}{\tan A} \) or \( \frac{\cos A}{\sin A} \)
💡 Memory Trick!

To remember which goes with which: The "S" (Secant) goes with the "C" (Cosine), and the "C" (Cosecant) goes with the "S" (Sine). They always swap their starting letters!

Angles of Any Magnitude

In A-Math, angles aren't just in triangles; they can spin around a circle forever! We use the ASTC diagram (or the "CAST" rule) to know which function is positive in which quadrant:

  • Quadrant 1 (0° to 90°): All are positive.
  • Quadrant 2 (90° to 180°): Sine only is positive.
  • Quadrant 3 (180° to 270°): Tangent only is positive.
  • Quadrant 4 (270° to 360°): Cosine only is positive.

Analogy: Think of it as "Add Sugar To Coffee"!

Quick Review: Special Angles

You must know the exact values for \(30^\circ, 45^\circ,\) and \(60^\circ\) (or \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \) in radians). For example, \( \sin 45^\circ = \frac{1}{\sqrt{2}} \) and \( \tan 60^\circ = \sqrt{3} \). Keep a reference table handy until you've memorized them!

Key Takeaway: There are six functions in total, and their signs (+ or -) depend on which quadrant the angle falls into.

2. Graphs: The Wavy World

Trigonometric functions look like repeating waves when graphed. The general forms are \( y = a \sin(bx) + c \) and \( y = a \cos(bx) + c \).

What do the letters mean?

  • a (Amplitude): This is the height of the wave from the middle. If \( a = 3 \), the wave goes 3 units up and 3 units down.
  • b (Periodicity): This tells you how many cycles happen in \( 360^\circ \) (or \( 2\pi \)). The Period (the length of one full wave) is calculated as \( \frac{360^\circ}{b} \) or \( \frac{2\pi}{b} \).
  • c (Vertical Shift): This moves the entire wave up or down.
Did you know?

The Tangent graph is different! It doesn't have an amplitude because it goes up to infinity, and it has "fences" called asymptotes where the graph cannot exist.

Key Takeaway: Amplitude is height, Period is length, and graphs repeat themselves regularly.

3. Trigonometric Identities: Your Toolbelt

Identities are equations that are always true. You use them to simplify messy expressions or prove that two sides of an equation are equal.

The "Big Three" Pythagorean Identities:

  1. \( \sin^2 A + \cos^2 A = 1 \)
  2. \( \sec^2 A = 1 + \tan^2 A \)
  3. \( \text{cosec}^2 A = 1 + \cot^2 A \)

Addition & Double Angle Formulae:

These help you break down angles like \( (A + B) \) or \( (2A) \):

  • \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
  • \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \) (Notice the sign flip!)
  • \( \sin 2A = 2 \sin A \cos A \)
  • \( \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A \)
Common Mistake to Avoid:

Students often think \( \sin(2A) \) is the same as \( 2\sin(A) \). It’s not! Always use the formula. For example, if \( A = 30^\circ \), \( \sin(60^\circ) \neq 2\sin(30^\circ) \).

Key Takeaway: Identities are like shortcuts. If you see \( \sin^2 A + \cos^2 A \), you can immediately turn it into a 1!

4. The R-Formula

Sometimes we have a mix of sine and cosine, like \( a \cos \theta + b \sin \theta \). This is hard to solve, so we turn it into one single wave:

\( R \cos(\theta \pm \alpha) \) or \( R \sin(\theta \pm \alpha) \)

To find R: \( R = \sqrt{a^2 + b^2} \)
To find \( \alpha \): \( \tan \alpha = \frac{opposite\_coefficient}{adjacent\_coefficient} \)

Key Takeaway: The R-formula is used to find the maximum/minimum value of an expression (the maximum is just \( R \), the minimum is \( -R \)).

5. Solving Trigonometric Equations

Solving \( \sin x = 0.5 \) is like a 3-step dance:

  1. Step 1: Find the Basic Angle (\( \alpha \)). Ignore the negative sign if there is one. \( \alpha = \sin^{-1}(0.5) = 30^\circ \).
  2. Step 2: Identify the Quadrants. Look at the sign (+ or -) and the ASTC diagram. Since sine is positive, we look at Quadrants 1 and 2.
  3. Step 3: Find the angles within the range.
    • Q1: \( x = \alpha = 30^\circ \)
    • Q2: \( x = 180^\circ - \alpha = 150^\circ \)
Quick Review: Inverse Trig Limits

When you use your calculator for \( \sin^{-1}, \cos^{-1}, \text{ or } \tan^{-1} \), it only gives you Principal Values (usually the smallest angle). You must use the quadrants to find the other possible answers!

Key Takeaway: Always check your range! If the question asks for \( 0^\circ \le x \le 360^\circ \), make sure you didn't miss any "spin" around the circle.

6. Proving Identities

Proving looks scary, but it’s just a game of substitution. Here are some tips:

  • Start with the more complex side (usually the Left-Hand Side).
  • Change everything to Sine and Cosine if you get stuck.
  • Look for fractions—try to combine them using a common denominator.
  • Look for squares—these are hints to use the Pythagorean identities.

"Don't worry if it takes a few tries. Proving is like solving a puzzle; sometimes you have to try a different piece to see if it fits!"

Key Takeaway: Never move things across the equals sign in a proof. Work on one side until it looks exactly like the other!