How to draw trig graphs and read off max, min and solutions on the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II
On the fx-CG50: MENU → 5 (GRAPH) … SHIFT F5 (G-SOLVE).On the TI-84 Plus CE: 2nd trace (CALC).
Graph y = a sin(bx) + d in the right angle mode, read off the maximum, minimum and period, and solve trig equations in an interval. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
Checking a sketch of a transformed sin, cos or tan graph, the number of solutions in an interval, and the solutions of a trig equation before you write the method. Degrees or radians depends on the interval in the question.
Practise it: Trigonometry · IAL P1 trigonometry.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Trig graphs and angle mode on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- Angle mode: SHIFT MENU (SET UP), move down to Angle, choose Rad (Deg for a question in degrees), EXIT.
- MENU 5 (GRAPH). Y1 = 3 sin(2X) + 1 EXE, Y2 = 2 EXE.
- SHIFT F3 (V-WINDOW): F2 (TRIG) gives a trig window for the angle mode, or type Xmin 0, Xmax π, Ymin −3, Ymax 5. EXIT, then F6 (DRAW).
- SHIFT F5 (G-SOLVE): F2 (MAX) and F3 (MIN) for the turning points; F5 (INTSECT) for where Y1 meets Y2, and ► for the next one.
- The angle unit shows in the status bar (Rad or Deg). A flat or squashed curve usually means the wrong mode.
Trig graphs and angle mode on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- mode: choose RADIAN (DEGREE for a question in degrees), enter.
- y=: Y1 = 3sin(2X) + 1, Y2 = 2.
- zoom 7: ZTrig, or window: Xmin 0, Xmax π, Ymin −3, Ymax 5, then graph.
- 2nd trace (CALC): 4: maximum, 3: minimum, 5: intersect. Answer Left Bound?, Right Bound? and Guess? (for intersect: first curve, second curve, Guess?).
- The status bar shows RADIAN or DEGREE: check it before the first trig question.
Trig graphs and angle mode on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Angle mode: home/on → Settings → Document Settings → Angle: Radian (or Degree).
- Add a Graphs page: f1(x) = 3·sin(2x) + 1, f2(x) = 2.
- menu → Window/Zoom (menu 4) → Window Settings: x from 0 to π, y from −3 to 5.
- menu → Analyze Graph → Maximum, Minimum or Intersection, then click a lower and an upper bound either side of the point.
Trig graphs and angle mode on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
The fx-991EX cannot draw graphs. A table of values in the right angle mode shows the shape, the maximum and minimum, and where the curve crosses a line; SOLVE then finds each solution.
- Angle mode: SHIFT MENU (SETUP) → Angle Unit → Radian (Degree for a question in degrees).
- MENU 9 (Table): f(x) = 3 sin(2X) + 1, =. Start 0, End π, Step π ÷ 12 (π is SHIFT ×10ˣ), =.
- Read the table: y is largest (4) at x = 0.785 and smallest (−2) at x = 2.356, and it passes 2 between the rows either side of each solution.
- Solve: MENU 1, type 3 sin(2X) + 1 = 2 (= is ALPHA CALC), SHIFT CALC (SOLVE), start value 0.2 =; again with start value 1.4.
Trig graphs and angle mode on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
The fx-991CW cannot draw graphs. A table of values in the right angle mode shows the shape, the maximum and minimum, and where the curve crosses a line.
- Angle mode: SETTINGS → Calc Settings → Angle Unit → Radian (Degree for a question in degrees).
- HOME → Table. Define f(x) = 3 sin(2x) + 1, then Table Range: Start 0, End π, Step π ÷ 12, and EXE.
- Show the table: y is largest (4) at x = 0.785 and smallest (−2) at x = 2.356.
- For the solutions use the Equation app's Solver with a starting value near each one (see Solving equations).
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Worked example
y = 3 sin(2x) + 1 for 0 ≤ x ≤ π (x in radians). Find (a) the maximum and minimum points, (b) the solutions of 3 sin(2x) + 1 = 2.
Answers (checked with Python)
(a) Maximum (0.785, 4) at x = π⁄4; minimum (2.36, −2) at x = 3π⁄4. The amplitude is 3 and the period is π. (b) x = 0.170 and x = 1.40.
Common calculator mistakes
- Degree mode for a radian question (or the other way round): the graph looks like a flat line or a dense zigzag, and every value is wrong.
- Not using the interval: the calculator shows every solution on screen, the question wants only those in 0 ≤ x ≤ π (or 0° ≤ x ≤ 360°).
- Period of sin(bx): it is 2π ÷ b (360° ÷ b), not 2π × b.
- Missing brackets: sin 2x + 1 typed as sin(2x + 1) moves the graph instead of the line.
- Giving decimals when the question says exact: π⁄4 and 3π⁄4, not 0.785 and 2.36.
If you get an error
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:INVALID DIM, ERR:SYNTAX
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Exponential and power models need positive data. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:INVALID DIM TI's knowledge base: one or more stat plots are turned on, but the lists they use are empty. Check: A stat plot left on from a statistics question, so the error appears when you graph a function. Fix: Turn off the stat plots you are not using, or put the data back in the lists.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
fx-991EX: Math ERROR, Syntax ERROR, Can't Solve
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation. In Matrix and Vector calculations it also appears when the sizes do not suit the operation. Check: A missing or extra bracket, or two operation signs in a row. With matrices or vectors: adding a 2 × 2 to a 3 × 3, or a product whose sizes do not match. Fix: Casio's advice: make the corrections. For matrices and vectors, check the size you set for each one.
Can't Solve SOLVE could not find a solution. Check: A typing slip in the equation, or a start value far from the root you want. Fix: Casio's advice: check the equation for mistakes, then give a start value close to the solution you expect.
fx-991CW: Math ERROR, Syntax ERROR
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation: it is typed in a form the calculator cannot read. Check: A missing or extra bracket, two operation signs in a row, or a function with nothing after it. Fix: Casio's advice: make the corrections and try again.
Do it in Desmos
Not in the exam. Desmos is not allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- Desmos works in radians unless you switch to degrees with the wrench (Graph Settings).
- Type y = 3sin(2x) + 1 and y = 2, and restrict the curve with {0 ≤ x ≤ π}.
- Click the curve for the maximum and minimum, and the crossings for the solutions.
Open desmos.com/calculator (free, in any browser) and type:
y=3sin(2x)+1{0≤ x≤pi}y=2
Questions
Can I use a calculator for trig graphs and angle mode in the exam?
Yes. Every UK A Level Maths paper (Edexcel 9MA0, AQA 7357, OCR H240) and every Edexcel IAL unit, P1 included, allows a calculator that has no CAS and no wireless link. The fx-CG50 qualifies.
What are the fx-CG50 keystrokes for trig graphs and angle mode?
Angle mode: SHIFT MENU (SET UP), move down to Angle, choose Rad (Deg for a question in degrees), EXIT. Then MENU 5 (GRAPH). Y1 = 3 sin(2X) + 1 EXE, Y2 = 2 EXE. Then SHIFT F3 (V-WINDOW): F2 (TRIG) gives a trig window for the angle mode, or type Xmin 0, Xmax π, Ymin −3, Ymax 5. EXIT, then F6 (DRAW). The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-CG50, TI-84 Plus CE, TI-Nspire CX II, Casio fx-991EX ClassWiz and Casio fx-991CW ClassWiz. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer to the accuracy the question asks for (often 3 significant figures). Keep full accuracy in later parts.
More calculator skills
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Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the models shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.