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Trigonometric graphs and equations
Past 90 degrees the triangle definitions give out and the unit circle takes over. Sine and cosine become coordinates and their graphs become waves. A calculator returns a principal value, so use periodicity, symmetry and the stated interval to find all valid solutions.
Builds on Triangles and the sine and cosine rules.
IN THIS TOPIC
- Sketch sine, cosine and tangent, and use their symmetry and period.
- Quote exact values at the standard angles, and extend them past 90° by symmetry.
- Use tan θ = sin θ/cos θ and sin²θ + cos²θ = 1 to reshape an equation.
- Find every solution in a stated interval, multiple angles and hidden quadratics included.
COMMON MISCONCEPTION
To solve sin x cos x = cos x, divide both sides by cos x.
Sine and cosine as coordinates
For angles beyond a triangle's reach, sine and cosine are defined by a circle. Measure an angle x anticlockwise from the positive horizontal axis, mark where it meets a circle of radius 1, and that point is (cos x, sin x). Below 90° this agrees with the triangle definitions. Above 90° it keeps going, around the circle and around again, so both graphs repeat every 360° and both are trapped between −1 and 1. Tangent is the odd one out. Being sin/cos it repeats every 180°, and it blows up wherever cos x = 0.
Exact values worth learning
Two triangles generate all the exact values you need. Halve a unit square for 45°, halve an equilateral triangle for 30° and 60°.
| x | sin x | cos x | tan x |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
| 180° | 0 | −1 | 0 |
Past 90° the graphs do the extending for you. Sine is symmetric about 90°, so sin 150° = sin 30° = ½. Cosine is symmetric about 180° and negative through the second and third quadrants, so cos 240° = −cos 60° = −½. Learn six values. Read the rest off the shape.
Two identities
Two identities do most of the algebraic work in this topic and neither appears in the booklet,
The first converts a mixed sin-and-cos equation into a tan equation. The second is the unit circle's Pythagoras, and it lets you replace sin² by cos², or the reverse, whenever a question turns out to be a quadratic underneath.
WORKED EXAMPLE
The equation you must not divide
Solve sin x cos x = cos x for 0 ≤ x < 360°.
Resist dividing both sides by cos x. Gather everything on one side and factorise: cos x (sin x − 1) = 0.
cos x = 0 gives x = 90° and 270°. sin x = 1 gives x = 90° again.
Solutions: x = 90° and 270°.
Dividing through by cos x would have thrown 270° away, because at 270° you would have been dividing by zero. Factorising never loses a solution. Dividing by an expression that can vanish always might.
Every solution in the interval
Your calculator's inverse button returns one principal value. The graph supplies the rest. For sine a second solution sits at 180° minus the principal value, and adding or subtracting whole periods generates the others.
When the angle inside the function is not simply x, use the same routine every time. Substitute θ for the whole bracket, widen the interval to whatever θ now runs over, solve there, and translate back at the end.
WORKED EXAMPLE
A shifted angle
Solve sin (x + 70°) = 0.5 for 0 < x < 360°.
Let θ = x + 70°. As x runs over its interval, θ runs from 70° to 430°.
sin θ = 0.5 at θ = 30°, 150°, 390°, … and inside (70°, 430°) that leaves θ = 150° or 390°.
Translate back. x = 80° or 320°.
The 30° root fell outside the widened interval and was discarded. The 390° root, one full wave later, lies inside it and becomes 320°. Widening the interval first is what makes both of those decisions reliable.
GUIDED PRACTICE
A doubled angle
Solve 3 + 5 cos 2x = 1 for −180° < x < 180°, before opening the working.
Show the working
Rearranging gives cos 2x = −2/5. Let θ = 2x, so θ runs over (−360°, 360°).
The principal value is cos−1(−0.4) = 113.6°, and cosine's symmetry supplies θ = ±113.6° and ±246.4°.
Halving gives x = ±56.8° and ±123.2°. Four solutions.
Doubling the angle doubled the solution count, since 2x sweeps two full waves while x sweeps one. Solving inside the x-interval and finding only two answers is the standard error here.
INDEPENDENT PRACTICE
A hidden quadratic, trig edition
Solve 6 cos2 x + sin x − 5 = 0 for 0 ≤ x < 360°.
Show the working
Trade cos² for sin². 6(1 − sin2 x) + sin x − 5 = 0, which tidies to 6 sin2 x − sin x − 1 = 0.
Factorise with s = sin x: (3s + 1)(2s − 1) = 0, so sin x = 1/2 or −1/3.
sin x = 1/2 gives x = 30°, 150°. sin x = −1/3 has principal value −19.5°, which lands in range as x = 199.5° and 340.5°.
Four solutions: 30°, 150°, 199.5°, 340.5°. The identity reduced two trig ratios to one, and the rest is the substitution routine from the algebra unit.
ASSESSMENT FOCUS
- Sketch the relevant graph across the widened interval before collecting anything. Count the crossings, then go and find them.
- For sin the partner solution is 180° − θ. For cos it is −θ, or equivalently 360° − θ. For tan, add 180°. Name the rule you are using.
- Substitute θ for the whole bracket, widen the interval to match, solve there, translate back. More marks go missing at the widening step than anywhere else in this topic.
- Never divide by cos x, sin x or anything else that can be zero. Move it all to one side and factorise.
- Exact-value questions want the surd. Write sin 60° = √3/2, not 0.866.
- Produce as many answers as your sketch predicted. Three crossings and two answers means one is still missing.
CHECK YOURSELF
Solve tan x = √3 for 0 ≤ x < 360°, giving exact answers.
Show a hint
One exact value, then tangent's period.
Show the answer
The exact-value table gives the principal value x = 60°.
Tangent repeats every 180°, so the other solution in range is 60° + 180° = 240°.
Solutions: x = 60° and 240°. The graph shows two branches crossing √3 in the interval, which is the count you were expecting.
The unit circle defines sin and cos for every angle; the graphs repeat and the table of exact values covers the rest.
Widen the interval with the substituted angle, find every crossing in it, then translate back.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the trigonometric graphs and equations questions page.
CHECK YOUR PROGRESS
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- Sketch sine, cosine and tangent, and use their symmetry and period.
- Quote exact values at the standard angles, and extend them past 90° by symmetry.
- Use tan θ = sin θ/cos θ and sin²θ + cos²θ = 1 to reshape an equation.
- Find every solution in a stated interval, multiple angles and hidden quadratics included.
Open the full revision checklist to see every objective in the course in one place.