Extending Trigonometry Beyond 90 Degrees Revision Notes | GCSE AQA Higher Further Maths

1. Why Trigonometry Needs to Go Beyond 90°

You already know how to find $\sin$, $\cos$, and $\tan$ for angles in a right-angled triangle — but triangles only give you angles between 0° and 90°. Real problems in maths and physics involve angles like 130°, 250°, or 310°, and your calculator handles these without complaint. The question is: do you know what is actually happening?

This lesson extends the definitions of $\sin \theta$, $\cos \theta$, and $\tan \theta$ to cover any positive angle up to 360°, using the coordinate plane rather than a triangle. Once you see the pattern, you will be able to find every angle that matches a given trig value — which is exactly what exam questions demand.

2. The Unit Circle: Redefining Sin, Cos, and Tan

Picture a circle of radius 1 centred at the origin. A point P moves around this circle, starting at $(1, 0)$ and travelling anticlockwise — that direction is the positive direction for measuring angles. The angle $\theta$ is always measured from the positive x-axis to the line OP.

This is the key shift: instead of opposite-over-hypotenuse, you are reading off coordinates. The x-coordinate of P gives $\cos \theta$ and the y-coordinate gives $\sin \theta$. Because coordinates can be negative, trig values can now be negative too — and that is the whole point.

3. The Four Quadrants and Signs of Trig Values

As $\theta$ increases from 0° to 360°, the point P passes through four quadrants. The sign of each trig value depends entirely on whether the x- and y-coordinates of P are positive or negative in that quadrant.

A common memory aid is CAST, read anticlockwise starting from Quadrant IV: Cosine positive (Q IV), All positive (Q I), Sine positive (Q II), Tangent positive (Q III). You do not need to memorise the acronym — understanding the coordinate signs is more reliable — but many students find it a useful shortcut.

4. Reference Angles: The Bridge to Familiar Values

Here is the key insight that makes every calculation manageable. For any angle $\theta$ in Quadrants II, III, or IV, there is a corresponding acute angle — called the reference angle — that has the same trig magnitude. You find the value using the reference angle, then apply the correct sign for the quadrant.

Worked example: Finding sin 150°

Find the exact value of $\sin 150°$.

1. $150°$ lies between $90°$ and $180°$, so it is in Quadrant II. — In Q II, sine is positive.
2. Reference angle $= 180° - 150° = 30°$ — Use the Q II rule: subtract from 180°. [M1]
3. $\sin 30° = \dfrac{1}{2}$ — This is a standard exact value.
4. $\sin 150° = +\dfrac{1}{2}$ — Sine is positive in Q II, so the sign is positive. [A1]

Answer: $\sin 150° = \dfrac{1}{2}$

Worked example: Finding cos 240°

Find the exact value of $\cos 240°$.

1. $240°$ lies between $180°$ and $270°$, so it is in Quadrant III. — In Q III, cosine is negative.
2. Reference angle $= 240° - 180° = 60°$ — Use the Q III rule: subtract 180°. [M1]
3. $\cos 60° = \dfrac{1}{2}$ — Standard exact value.
4. $\cos 240° = -\dfrac{1}{2}$ — Cosine is negative in Q III. [A1]

Answer: $\cos 240° = -\dfrac{1}{2}$

5. Finding All Angles for a Given Trig Value

Exam questions often ask you to find all angles between 0° and 360° that satisfy something like $\sin \theta = 0.5$. There are typically two such angles (sometimes one, at a boundary). The method is always the same: find the reference angle first, then use the quadrant rules to locate both solutions.

The sign of the given value tells you which quadrants to look in. A positive sine means Q I and Q II. A negative cosine means Q II and Q III. A positive tangent means Q I and Q III. Once you know the quadrants, you build each angle from the reference angle using the rules from Section 4.

Worked example: Solving sin θ = 0.5 for 0° ≤ θ ≤ 360°

Find all values of $\theta$ in the range $0° \leq \theta \leq 360°$ such that $\sin \theta = 0.5$.

1. $\sin \theta = 0.5 > 0$ — Sine is positive in Quadrants I and II.
2. $\theta_{\text{ref}} = \sin^{-1}(0.5) = 30°$ — Use the inverse sine of the positive value. [M1]
3. $\theta_1 = 30°$ — In Q I, the angle equals the reference angle directly.
4. $\theta_2 = 180° - 30° = 150°$ — In Q II, subtract the reference angle from 180°. [M1]
5. $\theta = 30°$ or $\theta = 150°$ — Both satisfy $\sin \theta = 0.5$ in the given range. [A1]

Answer: $\theta = 30°$ or $\theta = 150°$

Worked example: Solving cos θ = −0.5 for 0° ≤ θ ≤ 360°

Find all values of $\theta$ in the range $0° \leq \theta \leq 360°$ such that $\cos \theta = -0.5$.

1. $\cos \theta = -0.5 < 0$ — Cosine is negative in Quadrants II and III.
2. $\theta_{\text{ref}} = \cos^{-1}(0.5) = 60°$ — Use the inverse cosine of the positive version of the value. [M1]
3. $\theta_1 = 180° - 60° = 120°$ — In Q II, the angle is $180°$ minus the reference angle.
4. $\theta_2 = 180° + 60° = 240°$ — In Q III, the angle is $180°$ plus the reference angle. [M1]
5. $\theta = 120°$ or $\theta = 240°$ — Both satisfy $\cos \theta = -0.5$ in the given range. [A1]

Answer: $\theta = 120°$ or $\theta = 240°$

The graph above shows $y = \sin \theta$ from 0° to 360° alongside the horizontal line $y = 0.5$. The two intersection points confirm $\theta = 30°$ and $\theta = 150°$ — exactly what the algebra gave. Whenever you solve a trig equation, sketching the graph is a powerful sanity check.

6. Practice: Putting It All Together

The method is now complete. Before looking at exam traps, work through these problems independently. For each one, identify the quadrant(s), find the reference angle, and apply the sign rule.

7. Exam Technique and Common Mistakes

This topic has a small number of very predictable traps. Examiners set questions specifically to catch students who apply the right method but forget the sign, or who find only one solution when two are required. The guidance below addresses the most common errors seen in mark schemes.

Every spec requirement for this topic comes down to two skills working together: knowing the sign of each trig function in each quadrant, and converting between an angle and its reference angle. Master those two tools and you can handle any question on $\sin \theta$, $\cos \theta$, and $\tan \theta$ up to 360°.