Geometry I (AQA GCSE Further Maths): Revision Notes

Solution of trigonometric equations

The tangent graph

Understanding the tangent function is essential for solving trigonometric equations. The tangent of an angle can be calculated using the relationship $\tan \theta = \frac{y}{x}$, or alternatively using the ratio $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Understanding asymptotes

The tangent function has some unique characteristics that make it different from sine and cosine. Most importantly, the tangent function becomes undefined at certain angles. This occurs when $\theta = 90°, 270°$, and similar values where the cosine equals zero (since we'd be dividing by zero).

When we look at the graph of the tangent function, we can see vertical dashed lines at these problematic angles. These lines are called asymptotes - they represent values that the function approaches but never actually reaches. The function shoots up to positive infinity on one side and drops to negative infinity on the other side of each asymptote.

Periodic nature of tangent

Just like sine and cosine, the tangent function is periodic, meaning it repeats its pattern at regular intervals. However, while sine and cosine have a period of 360°, the tangent function repeats every 180°. This shorter period is crucial when finding all solutions to tangent equations.

Finding solutions to trigonometric equations

When solving equations like $\sin \theta = 0.5$, your calculator will give you one answer - but this is just the beginning. Trigonometric equations typically have multiple solutions within any given range.

Using your calculator effectively

Understanding principal values

The value your calculator gives you (like 30° for $\sin \theta = 0.5$) is called the principal value. This is just one solution, and there are specific ranges where calculators give principal values:

• For cosine: between 0° and 180°
• For sine: between -90° and 90°
• For tangent: between -90° and 90°

Finding all solutions

Let's examine how many solutions exist by looking at graphs. When we plot $y = \sin \theta$ and draw a horizontal line at $y = 0.5$, we can see multiple intersection points.

The graph shows that $\sin \theta = 0.5$ has solutions at 30° and 150° within one complete cycle, and this pattern repeats every 360°. So all solutions would be: $\theta = ..., -330°, -210°, 30°, 150°, 390°, 510°, ...$

Worked examples

Systematic method for solving trigonometric equations