Trigonometric Graphs and Equations

Your calculator gives one answer to a trigonometric equation, but the graphs repeat — so there are usually more solutions in the range. This topic is about finding the ones the calculator doesn’t tell you about.

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1. The three graphs

y = sin x

• Starts at (0, 0), rises to 1 at 90°, back to 0 at 180°, down to −1 at 270°, back to 0 at 360°
• Range: −1 ≤ sin x ≤ 1
• Period: 360° — it repeats every 360°
• Amplitude: 1

y = cos x

• Starts at (0, 1), falls to 0 at 90°, −1 at 180°, 0 at 270°, back to 1 at 360°
• Range: −1 ≤ cos x ≤ 1
• Period: 360°, amplitude 1

The cosine graph is the sine graph shifted 90° to the left. They have the same shape, just a different starting point — which is why sin and cos values swap around between 0° and 90°.

y = tan x

• Passes through (0, 0), rises steeply, with asymptotes at 90° and 270°
• Range: all values — unlike sin and cos, it is not limited to −1 to 1
• Period: 180° — it repeats twice as often

sin and cos never exceed 1. If you get sin x = 1.5, there is no solution — check for an arithmetic error. tan has no such limit.

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2. Key values

x	0°	30°	45°	60°	90°	180°	270°	360°
sin x	0	1/2	√2/2	√3/2	1	0	−1	0
cos x	1	√3/2	√2/2	1/2	0	−1	0	1
tan x	0	1/√3	1	√3	—	0	—	0

sin 60° = √3/2 ≈ 0.866, and cos 60° = 1/2. Mixing up which is which at specific angles was recorded repeatedly — the table is worth memorising.

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3. Solving trigonometric equations

Your calculator gives the principal value — one solution. To find the rest within a given range, use the symmetry of the graph.

For sin x = k, the second solution is 180° − x. For cos x = k, the second solution is 360° − x. For tan x = k, add 180°.

Example: solve sin x = 0.5 for 0° ≤ x ≤ 360°

1. Calculator: x = sin⁻¹(0.5) = 30°
2. Second solution: 180 − 30 = 150°
3. x = 30° or 150°

Example: solve cos x = 0.5 for 0° ≤ x ≤ 360°

1. Calculator: x = 60°
2. Second solution: 360 − 60 = 300°
3. x = 60° or 300°

Example with a negative value: solve sin x = −0.5 for 0° ≤ x ≤ 360°

1. Calculator: sin⁻¹(−0.5) = −30° — outside the range
2. Using symmetry: 180 − (−30) = 210°, and 360 + (−30) = 330°
3. x = 210° or 330°

A negative value pushes the solutions into the third and fourth quadrants — where sine is negative. The calculator’s negative answer is correct but out of range, so use it with the symmetry rules rather than discarding it.

Always check the range given in the question and give every solution inside it. Stopping at the calculator’s answer loses at least half the marks.

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4. The quadrant (CAST) rule

Which functions are positive in each quadrant:

Quadrant	Angles	Positive
1st	0°–90°	All
2nd	90°–180°	Sine only
3rd	180°–270°	Tangent only
4th	270°–360°	Cosine only

Remembered as CAST, read anticlockwise from the fourth quadrant: Cos, All, Sin, Tan.

Use the quadrants to check your answers rather than as the primary method. Sketching the graph and using the symmetry rules is more reliable, and tutors found students consistently muddled the CAST diagram — knowing which quadrants make sine negative matters more than the mnemonic itself.

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5. Reading and sketching graphs

Sketch method: mark the key points (0°, 90°, 180°, 270°, 360°) and join with a smooth curve.

Solving graphically: to solve sin x = 0.5, draw the horizontal line y = 0.5 and read where it crosses the curve. This shows visually why there are two solutions in 0°–360°, and it is a good check on the algebra.

Transformations (for the Extended tier):

Function	Effect
y = a sin x	amplitude becomes a
y = sin x + c	shifts up by c
y = sin(x + c)	shifts left by c

A number inside the bracket shifts horizontally; a number outside shifts vertically or stretches. Confusing a shift with a stretch was a recorded error — compare a few coordinates on both graphs to decide which has happened.

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6. Mistakes that cost marks

Giving only the calculator’s answer.

Using 180 − x for cosine (it is 360 − x).

Using 360 − x for sine (it is 180 − x).

Missing solutions inside the range, or giving some outside it.

Accepting sin x > 1 as possible.

Muddling the CAST quadrants.

Confusing sin and cos values at key angles.

Forgetting tan repeats every 180°, not 360°.

Leaving the calculator in radian mode.

Confusing a horizontal shift with a stretch.

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Frequently asked questions

What is the period of the sine graph? 360°. Cosine is also 360°; tan is 180°.

What is the range of sin x? −1 to 1. Same for cos. tan has no limit.

Why does my equation have two answers? Because the graph repeats and is symmetrical, so a horizontal line crosses it more than once.

How do I find the second solution for sine? 180° − x.

How do I find the second solution for cosine? 360° − x.

What about tangent? Add 180°.

What is the CAST rule? Which functions are positive in each quadrant: All, Sine, Tangent, Cosine.

What if sin x = 1.5? No solution — sine never exceeds 1.

How is the cosine graph related to the sine graph? It is the sine graph shifted 90° to the left.

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Quick revision checklist

• I can sketch y = sin x, y = cos x and y = tan x
• I know their periods, ranges and amplitudes
• I know tan repeats every 180° and has asymptotes
• I know sin and cos never exceed 1
• I know the exact values at 0°, 30°, 45°, 60° and 90°
• I can find the second solution for sine (180 − x)
• I can find the second solution for cosine (360 − x)
• I can handle negative values
• I give all solutions in the stated range
• I can use the quadrant rule as a check
• I can solve graphically by drawing a horizontal line
• I can identify amplitude changes and shifts
• My calculator is in degree mode

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These notes cover trigonometric graphs and equations in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.