01 · Sine landmarks
List the zeros of sin θ on −360° ≤ θ ≤ 360°.
Hint
Zeros are integer multiples of 180°.
Worked solution
−360°, −180°, 0°, 180°, 360°
Understand · explore · practise
Sine, cosine and tangent graphs
Sketch sine, cosine and tangent in degrees. Learn exact landmarks, periods, negative angles, tangent asymptotes and how to read repeated solutions.
Before you startDegree angles, coordinates and basic trigonometric ratios
01 / Beyond right triangles
A unit circle extends sine and cosine to every angle.
Measure θ anticlockwise from the positive horizontal axis on a circle of radius 1. The point has coordinates (cos θ, sin θ): cosine is its horizontal coordinate and sine its vertical coordinate.
A clockwise rotation gives a negative angle. One complete turn adds 360° and returns to the same point. This explains why sine and cosine repeat every 360°.
tan θ = sin θ / cos θ, when cos θ ≠ 0
The graph model uses degrees throughout. Choose a function and angle; at a tangent asymptote there is no finite function value to plot.
Explore the basic degree graphsChoose and compare
sin 30° = 0.5. Sine has period 360° and range [−1,1]. The green point marks the selected angle.
Watch the three graphs reveal their landmarks
Pause, replay or seek freely. The notes explain the same idea and stay in view.
02 / The sine graph
Start at zero and rise to one.
θ: 0°, 90°, 180°, 270°, 360°
sin θ: 0, 1, 0, −1, 0
Join these landmarks with a smooth wave. The range is [−1,1] and the fundamental period (smallest positive repeat) is 360°.
Zeros: θ = 180°n
Maxima: θ = 90° + 360°n
Minima: θ = 270° + 360°n
Here and below n is any integer. Select only the points inside the requested interval.
03 / The cosine graph
Start at one, not zero.
θ: 0°, 90°, 180°, 270°, 360°
cos θ: 1, 0, −1, 0, 1
Cosine has the same range and fundamental period as sine, but a different starting point.
Zeros: θ = 90° + 180°n
Maxima: θ = 360°n
Minima: θ = 180° + 360°n
04 / The tangent graph
Leave a gap at each vertical asymptote.
Tangent repeats every 180°. It is zero at 180°n and undefined at 90° + 180°n, because cosine is zero there.
tan 0° = 0
tan 45° = 1
tan(−45°) = −1
Each branch rises from arbitrarily negative values to arbitrarily positive values between consecutive asymptotes. Tangent has range ℝ and no maximum or minimum. A vertical asymptote is not part of the curve: never connect a line through the discontinuity.
05 / Exact values
Build a small set of dependable landmarks.
A 45°–45°–90° triangle with legs 1 has hypotenuse √2. Halving an equilateral triangle of side 2 gives a 30°–60°–90° triangle with sides 1, √3 and 2.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
06 / Negative and related angles
Read signs from the circle or the graphs.
sin(−θ) = −sin θ
cos(−θ) = cos θ
tan(−θ) = −tan θ
Sine and tangent have rotational symmetry about the origin; cosine has reflection symmetry in the vertical axis.
sin(180° − θ) = sin θ
cos(180° − θ) = −cos θ
tan(180° − θ) = −tan θ
Adding 180° reverses both circle coordinates: sine and cosine change sign, while tangent is unchanged. Subtracting θ from 360° keeps cosine but reverses sine and tangent. Tangent statements apply only where both sides are defined.
07 / Read repeated solutions
A horizontal level can meet several branches.
On −180° ≤ θ ≤ 360°, the line y = 1/2 meets the sine graph at 30° and 150°. The corresponding negative values obtained by subtracting 360° lie outside this interval.
For cos θ = 1/2 on the same interval, the solutions are −60°, 60° and 300°. For tan θ = 1, they are −135°, 45° and 225°.
Specify the interval and check its endpoints. A picture helps locate the answers, but use exact values and periods to justify them. Do not count an asymptote as an intersection.
08 / Your turn
Use landmarks, then extend by the correct period.
All angles are in degrees.
02 · Cosine extrema
Where does cos θ equal −1 on −360° ≤ θ ≤ 360°?
Hint
Start at 180° and repeat every 360°.
Worked solution
θ = −180° or 180°
03 · Tangent gaps
List the vertical asymptotes of tan θ on −180° < θ < 360°.
Hint
Use 90° + 180°n.
Worked solution
θ = −90°, 90°, 270°
04 · Negative angle
Find sin(−30°), cos(−60°) and tan(−45°).
Hint
Cosine is even; sine and tangent are odd.
Worked solution
−1/2, 1/2, −1
05 · Two sine values
Find θ where sin θ = −√3/2 on 0° ≤ θ ≤ 360°.
Hint
The reference angle is 60°; sine is negative below the horizontal axis.
Worked solution
θ = 240° or 300°
06 · Repeated tangent
Find θ where tan θ = √3 on −180° ≤ θ ≤ 360°.
Hint
Start at 60° and add or subtract 180°.
Worked solution
θ = −120°, 60°, 240°
07 · A false maximum
A graphing window shows tan θ only between −4 and 4. Is its maximum value 4?
Hint
A plotting boundary is not a function bound.
Worked solution
No. Tangent is unbounded above and below. The window clips the curve; it does not change the function’s range.
09 / Recap
Know the landmarks and respect the gaps.
- Sine starts at 0; cosine starts at 1.
- Sine and cosine repeat every 360° and stay between −1 and 1.
- Tangent repeats every 180° and has vertical asymptotes.
- Mark degree units and restrict answers to the stated interval.
- Use symmetry and periods to justify repeated values.
Section 1 of 9 · Beyond right triangles