CIE IGCSE Mathematics (0580) Trigonometric functions Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Trigonometric functions Study Notes
LEARNING OBJECTIVE
- Trigonometric Functions
Key Concepts:
- Trigonometric Functions
Trigonometric Functions
Trigonometric Functions
You should be able to recognise, sketch and interpret the graphs of the three main trigonometric functions:
- \( y = \sin x \)
- \( y = \cos x \)
- \( y = \tan x \)
These graphs should be known for values of \( x \) from \( 0^\circ \) to \( 360^\circ \). The angle \( x \) is measured in degrees.
1. Graph of \( y = \sin x \)
- Starts at 0
- Maximum at \( 90^\circ \) → \( y = 1 \)
- Back to 0 at \( 180^\circ \)
- Minimum at \( 270^\circ \) → \( y = -1 \)
- Back to 0 at \( 360^\circ \)
The sine graph is smooth, wave-shaped, and periodic with period 360°.
Example:
What is the value of \( \sin 180^\circ \)?
▶️ Answer/Explanation
From the graph or exact value table: \( \sin 180^\circ = 0 \)
2. Graph of \( y = \cos x \)
- Starts at 1
- 0 at \( 90^\circ \)
- Minimum at \( 180^\circ \) → \( y = -1 \)
- 0 at \( 270^\circ \)
- Back to 1 at \( 360^\circ \)
The cosine graph is also wave-shaped, periodic, and very similar to sine but shifted 90° to the left.
Example:
Find the value of \( \cos 270^\circ \).
▶️ Answer/Explanation
From the cosine graph: \( \cos 270^\circ = 0 \)
3. Graph of \( y = \tan x \)
- Passes through the origin (0, 0)
- Undefined at \( 90^\circ \) and \( 270^\circ \) → vertical asymptotes (dotted lines)
- Repeats every 180° (period = 180°)
The tangent graph increases steeply and has vertical asymptotes where \( \tan x \) is undefined.
Example:
What is the value of \( \tan 45^\circ \)? What happens at \( \tan 90^\circ \)?
▶️ Answer/Explanation
\( \tan 45^\circ = 1 \)
\( \tan 90^\circ \) is undefined → the graph has a vertical asymptote here.
Solving Trigonometric Equations
Solving Trigonometric Equations
You should be able to solve equations involving:
- \( \sin x = a \)
- \( \cos x = a \)
- \( \tan x = a \)
All solutions are found within the interval \( 0^\circ \leq x \leq 360^\circ \), using the unit circle and trigonometric graphs or CAST diagram.
Example:
Solve \( \sin x = \frac{1}{2} \) for \( 0^\circ \leq x \leq 360^\circ \).
▶️ Answer/Explanation
We know \( \sin x = \frac{1}{2} \) at \( x = 30^\circ \).
Sine is positive in Quadrants I and II (using CAST).
So, the two solutions are:
\( x = 30^\circ \) and \( x = 180^\circ – 30^\circ = \boxed{150^\circ} \)
Example:
Solve: \( \sin x = \dfrac{\sqrt{3}}{2} \) for \( 0^\circ \leq x \leq 360^\circ \)
▶️ Answer/Explanation
We know from exact values that:
\( \sin 60^\circ = \dfrac{\sqrt{3}}{2} \)
Sine is positive in Quadrants I and II (use CAST diagram).
So, the two solutions are:
\( x = 60^\circ \) and \( x = 180^\circ – 60^\circ = \boxed{120^\circ} \)
Example:
Solve: \( 2\cos x + 1 = 0 \) for \( 0^\circ \leq x \leq 360^\circ \)
▶️ Answer/Explanation
Rearrange the equation:
\( 2\cos x + 1 = 0 \Rightarrow \cos x = -\dfrac{1}{2} \)
Use exact value:
\( \cos 60^\circ = \dfrac{1}{2} \), so \( \cos x = -\dfrac{1}{2} \) has reference angle \( 60^\circ \)
Cosine is negative in Quadrants II and III
So the solutions are:
\( x = 180^\circ – 60^\circ = 120^\circ \) and \( x = 180^\circ + 60^\circ = \boxed{240^\circ} \)