International Mathematics (0607) | Trigonometric functions

👋 Welcome to Trigonometric Functions!

Hello there! Trigonometry might sound complex, but at its heart, it's just a fantastic tool for measuring things we can't physically reach, like the height of a mountain or the distance to a star. In this chapter, we are moving beyond simple right-angled triangles to explore how these relationships—sine, cosine, and tangent—behave as continuous functions, how to use them in any triangle, and how to apply them in three dimensions.

Don't worry if the graphs look like waves! We’ll break them down step-by-step. Let’s dive in and unlock the secrets of trigonometry!

Section 1: The Foundation – Right-Angled Triangles (E7.1 & E7.2)

Pythagoras and SOH CAH TOA

Before dealing with angles, remember the bedrock of triangle calculations: Pythagoras’ Theorem (E7.1).
For any right-angled triangle with hypotenuse \(c\): $$a^2 + b^2 = c^2$$

Trigonometry ratios rely on the position of the angle (\(\theta\)) you are working with:

• Hypotenuse (H): The side opposite the right angle (always the longest side).
• Opposite (O): The side directly across from the angle \(\theta\).
• Adjacent (A): The side next to the angle \(\theta\) (that is not the hypotenuse).

Memory Aid: SOH CAH TOA

This is the crucial mnemonic for remembering the three basic ratios (E7.2):

• SOH: Sine \(\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• CAH: Cosine \(\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• TOA: Tangent \(\theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Angles of Elevation and Depression

These concepts are used in 2D problems (E7.2.4) and rely on the idea of a horizontal line.

• Angle of Elevation: The angle measured upwards from the horizontal line of sight to an object above.
• Angle of Depression: The angle measured downwards from the horizontal line of sight to an object below.

Tip: The angle of elevation from person A to person B is always equal to the angle of depression from person B to person A (alternate angles on parallel horizontal lines).

Section 2: Exact Values and Trigonometric Graphs (E7.3 & E7.4)

Exact Trigonometric Values (E7.3)

In non-calculator exams (Paper 2), you must know the exact values for specific angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\). These often involve surds and must be given exactly (not as decimals).

The most common values to memorize:

• \(\sin 30^\circ = \frac{1}{2}\)
• \(\cos 60^\circ = \frac{1}{2}\)
• \(\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\) or \(\frac{1}{\sqrt{2}}\)
• \(\tan 45^\circ = 1\)
• \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)

Did you know? These values come from constructing a square and an equilateral triangle and using Pythagoras' theorem!

Trigonometric Functions and Graphs (E7.4)

When we plot the values of sin \(\theta\), cos \(\theta\), and tan \(\theta\) for all angles, we get continuous wave patterns. You need to be able to recognise, sketch, and interpret these graphs for the domain \(0^\circ \le x \le 360^\circ\).

1. The Sine Function: \(y = \sin x\)

• Shape: Starts at the origin (0, 0), rises to 1, drops to –1, and returns to 0.
• Key Points: Max at \((90^\circ, 1)\), Min at \((270^\circ, -1)\), Zeros at \(0^\circ, 180^\circ, 360^\circ\).
• Period: \(360^\circ\) (The wave repeats every \(360^\circ\)).
• Range: \(-1 \le y \le 1\).

2. The Cosine Function: \(y = \cos x\)

• Shape: Looks exactly like the sine wave, but shifted left by \(90^\circ\). It starts at its maximum height.
• Key Points: Max at \((0^\circ, 1)\) and \((360^\circ, 1)\), Min at \((180^\circ, -1)\). Zeros at \(90^\circ\) and \(270^\circ\).
• Period: \(360^\circ\).
• Range: \(-1 \le y \le 1\).

3. The Tangent Function: \(y = \tan x\)

The tangent function is very different because it is defined as \(\frac{\sin x}{\cos x}\). This means when \(\cos x = 0\), the function is undefined!

• Shape: Repeating 'S' curves.
• Period: \(180^\circ\) (It repeats twice as fast as sin and cos).
• Asymptotes: Vertical lines where the graph is undefined. These occur at \(90^\circ\) and \(270^\circ\). The graph tends towards these lines but never touches them.
• Zeros: \(0^\circ, 180^\circ, 360^\circ\).

Solving Trigonometric Equations (E7.4.2)

Solving equations like \(\sin x = 0.5\) means finding all the angles \(x\) in the range \(0^\circ \le x \le 360^\circ\) that satisfy the equation. Because the graphs are symmetrical waves, there are often two solutions.

Step-by-Step Method:

1. Find the Basic Angle (\(\alpha\)): Use the inverse trig function (e.g., \(x = \sin^{-1}(0.5)\)). This always gives you the acute (first quadrant) angle. Let's call this angle \(\alpha\).
2. Identify Quadrants/Symmetry: Look at the sign of the value you are solving for:
   • If \(\sin x\) is positive, solutions are \(\alpha\) and \((180^\circ - \alpha)\).
   • If \(\cos x\) is positive, solutions are \(\alpha\) and \((360^\circ - \alpha)\).
   • If \(\tan x\) is positive, solutions are \(\alpha\) and \((180^\circ + \alpha)\).
3. Find the remaining solutions: Apply the symmetry rules based on the positive or negative sign.

Example: Solve \(2 \cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).
First, rearrange: \(\cos x = -0.5\). Cosine is negative.
1. Find Basic Angle \(\alpha\): \(\cos^{-1}(0.5) = 60^\circ\). (Ignore the negative sign for this step).
2. Symmetry: Cosine is negative in the 2nd and 3rd quadrants.
3. Solutions:

• Solution 1 (2nd Q): \(180^\circ - 60^\circ = 120^\circ\)
• Solution 2 (3rd Q): \(180^\circ + 60^\circ = 240^\circ\)

The solutions are \(x = 120^\circ\) and \(x = 240^\circ\).

Section 3: Non-Right-Angled Triangles (E7.5)

The standard SOH CAH TOA rules only work for right-angled triangles. When you have a general triangle (any shape), you must use the Sine Rule, Cosine Rule, or Area Rule (E7.5).
(Remember: these formulas are provided in your Extended formula sheet!)

1. The Sine Rule

Used when you know:

• Two angles and any side (AAS or ASA)
• Two sides and a non-included angle (SSA – beware of the Ambiguous Case)

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

The Ambiguous Case: This only happens when finding an angle (Angle B) given two sides and a non-included angle (SSA). Since \(\sin \theta = \sin(180^\circ - \theta)\), your calculator might give you the acute angle, but an obtuse angle might also be valid. You must check if both angles (Acute and \(180^\circ - \text{Acute}\)) are geometrically possible in the triangle.

2. The Cosine Rule

Used when you know:

• Two sides and the included angle (SAS) – to find the third side.
• All three sides (SSS) – to find any angle.

To find a side \(a\): $$a^2 = b^2 + c^2 - 2bc \cos A$$

To find an angle \(A\): (You must rearrange the formula!) $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

3. Area of a Triangle

The area rule is used when you know two sides and the included angle (SAS) (E7.5.2). $$\text{Area} = \frac{1}{2}ab \sin C$$

Key Takeaway: Choose the right rule! Sine Rule needs pairs of sides/opposite angles. Cosine Rule is for SSS or SAS situations.

Section 4: Trigonometry in 3D (E7.6)

Trigonometry in three dimensions (3D) simply means applying Pythagoras' theorem and the SOH CAH TOA rules to shapes like cubes, pyramids, and prisms. The key challenge is identifying the correct right-angled triangle within the 3D figure.

The Key Skill: Finding the Angle Between a Line and a Plane

This is the most common type of 3D trig problem (E7.6).

Imagine a pencil (the Line) leaning against a table (the Plane). The angle between the pencil and the table surface is found by first imagining where the pencil's shadow falls directly beneath it.

Step-by-Step Procedure:

1. Identify the Line (L): The line segment whose angle you need to find (e.g., AG).
2. Identify the Plane (P): The surface it meets (e.g., the base ABCD).
3. Find the Projection (P'): The shadow of the line L onto the plane P. This is the line segment directly beneath L (e.g., AC is the projection of AG onto the base ABCD).
4. Form the Right-Angled Triangle: The triangle you need will be formed by the Line (L), the Projection (P'), and the vertical height (H) connecting the end of L to P'. (e.g., Triangle AGC, where C is the right angle if AG is the line and AC is the projection).
5. Solve: Use Pythagoras (to find the length of P' or L) and then SOH CAH TOA to find the required angle.

Analogy: Think of the line as a ramp. The angle of the ramp is measured between the ramp itself (the line) and the ground (its projection on the plane).

Final Check and Study Tips

Congratulations on completing the Trigonometric Functions chapter! This section links geometry and functions beautifully. Make sure you practice the following skills:

• Fluency in using the Sine and Cosine Rules, especially recognizing the Ambiguous Case.
• Instant recall of the exact trigonometric values for non-calculator papers.
• Visualizing and sketching the three main trigonometric graphs, knowing their zero points and maximum/minimum values.
• Confidently identifying the crucial right-angled triangle when solving 3D problems.

Keep practicing those problems—especially the 3D ones—and you will master this section!