Circle and trigonometric graphs
Circle graphs (Higher tier only)
A circle with centre (0, 0) and radius r has the equation:
x² + y² = r² (learn this)
It comes from Pythagoras: a point (x, y) on the circle is r away from the centre.
- To find the radius from an equation, square root the number on the right.
- To check whether a point is on the circle, substitute it and see if both sides are equal.
Example: the circle x² + y² = 25 has radius √25 = 5. Is the point (3, −4) on it?
- 3² + (−4)² = 9 + 16 = 25
- Yes, because it equals 25
Example: write the equation of the circle with centre (0, 0) and radius 6.
- r² = 6² = 36
- x² + y² = 36
Key terms
Tangents to a circle (Higher tier only)
A touches a circle at one point. It is perpendicular to the radius at that point.
- Find the gradient of the radius from (0, 0) to the point.
- The tangent's gradient is the negative reciprocal.
- Substitute the point into y = mx + c to find c.
Example: find the equation of the tangent to x² + y² = 25 at P(3, 4).
- Gradient of the radius = 4 ÷ 3 = 4/3
- Tangent gradient = −¾
- 4 = −¾ × 3 + c, so c = 4 + 9/4 = 25/4
- y = −¾x + 25/4 (or 4y + 3x = 25)
Key terms
Sine and cosine graphs (Higher tier only)
Both graphs are waves that repeat every 360°: their is 360°. Both stay between −1 and 1.
- y = sin x (curve A) starts at 0, peaks at 1 at 90°, and is −1 at 270°.
- y = cos x (curve B) starts at 1, is 0 at 90°, and is −1 at 180°.
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
| cos x | 1 | 0 | −1 | 0 | 1 |
Example: use the graph to find the values of x between 0° and 360° where cos x = 0.
- Curve B crosses the x-axis at 90° and 270°.
- x = 90° or x = 270°
Key terms
The tangent graph (Higher tier only)
y = tan x has a period of 180°, and its shape is different from sin and cos.
- tan 0° = 0, tan 45° = 1 and tan 180° = 0.
- tan x has no value at 90° and 270°: the graph shoots up and down towards these lines but never reaches them. They are .
Example: use the period to find tan 225°.
- tan repeats every 180°, so tan 225° = tan (225° − 180°) = tan 45°
- tan 225° = 1
Key terms
Solving trig equations (Higher tier only)
A calculator gives one solution. The graphs' symmetry gives the others between 0° and 360°.
| Equation | First solution | Second solution |
|---|---|---|
| sin x = k | sin⁻¹ k | 180° − first |
| cos x = k | cos⁻¹ k | 360° − first |
| tan x = k | tan⁻¹ k | first + 180° |
Example: solve sin x = 0.5 for 0° ≤ x ≤ 360°.
- sin⁻¹ 0.5 = 30°
- 180° − 30° = 150°
- x = 30° or x = 150°
Harder example: solve cos x = −0.5 for 0° ≤ x ≤ 360°.
- cos⁻¹ (−0.5) = 120°
- 360° − 120° = 240°
- x = 120° or x = 240°
Exam tip
Sketch the graph and draw the line y = k: it shows how many solutions there are and roughly where.