{"id":65228,"date":"2026-07-28T09:55:59","date_gmt":"2026-07-28T08:55:59","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=65228"},"modified":"2026-07-28T09:56:01","modified_gmt":"2026-07-28T08:56:01","slug":"math-30-1-formula-sheet","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/07\/28\/math-30-1-formula-sheet\/","title":{"rendered":"Math 30-1 Formula Sheet: Every Formula You Need for the Alberta Diploma Exam"},"content":{"rendered":"\n

This page contains the Math 30-1 formula sheet that Alberta students need for Math 30-1 and the Alberta Diploma Examination \u2014 organised by unit, explained where the formula requires context, and annotated with common exam mistakes to avoid.<\/p>\n\n\n\n

A formula sheet tells you what to use. It does not tell you when, how, or why. The difference between students who score well on the diploma exam and those who don’t is not which formulas they have memorised \u2014 it is whether they understand what those formulas mean and how to apply them in unfamiliar contexts. That understanding is what this guide is built to support, alongside the reference itself.<\/p>\n\n\n\n

For a full breakdown of the Math 30-1 curriculum, prerequisites, and diploma exam structure, see our Math 30-1 complete guide<\/a>.<\/p>\n\n\n\n

\"math<\/a><\/figure>\n\n\n\n
\n\n\n\n

What the Alberta Diploma Exam Provides vs What You Must Know<\/h2>\n\n\n\n

One of the most important things to understand before studying Math 30-1 formulas: the diploma exam does not provide a formula sheet<\/strong>. Students are expected to know every formula from memory.<\/p>\n\n\n\n

This is different from many school-based assessments where a reference sheet is permitted. On the diploma exam, if you cannot recall the double angle formula or the combination formula, you cannot use it. This makes deliberate memorisation \u2014 not just understanding \u2014 a necessary part of preparation.<\/p>\n\n\n\n

The formulas below are organised by unit. For each formula, the notation is standard Alberta curriculum notation. Where a formula has a common misapplication on the diploma exam, that is noted explicitly.<\/p>\n\n\n\n

\n\n\n\n

Math 30-1 Formula Sheet: Trigonometry Formulas<\/h2>\n\n\n\n

Trigonometry is the most heavily tested unit on the Math 30-1 diploma exam and contains the greatest number of formulas to know.<\/p>\n\n\n\n

Math 30-1 Formula Sheet: Fundamental Identities<\/h3>\n\n\n\n

Pythagorean Identities<\/strong>sin<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi>+<\/mo>cos<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi>=<\/mo>1<\/mn><\/mrow>\\sin^2\\theta + \\cos^2\\theta = 1<\/annotation><\/semantics><\/math>sin2\u03b8+cos2\u03b8=1 1<\/mn>+<\/mo>tan<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi>=<\/mo>sec<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi><\/mrow>1 + \\tan^2\\theta = \\sec^2\\theta<\/annotation><\/semantics><\/math>1+tan2\u03b8=sec2\u03b8 1<\/mn>+<\/mo>cot<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi>=<\/mo>csc<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>\u03b8<\/mi><\/mrow>1 + \\cot^2\\theta = \\csc^2\\theta<\/annotation><\/semantics><\/math>1+cot2\u03b8=csc2\u03b8<\/p>\n\n\n\n

From the first identity, derive the other two by dividing through by cos\u00b2\u03b8 and sin\u00b2\u03b8 respectively. Knowing the derivation means you never need to memorise all three separately.<\/em><\/p>\n\n\n\n

Reciprocal Identities<\/strong>csc<\/mi>\u2061<\/mo>\u03b8<\/mi>=<\/mo>1<\/mn>sin<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow><\/mfrac><\/mspace>sec<\/mi>\u2061<\/mo>\u03b8<\/mi>=<\/mo>1<\/mn>cos<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow><\/mfrac><\/mspace>cot<\/mi>\u2061<\/mo>\u03b8<\/mi>=<\/mo>1<\/mn>tan<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow>\\csc\\theta = \\frac{1}{\\sin\\theta} \\qquad \\sec\\theta = \\frac{1}{\\cos\\theta} \\qquad \\cot\\theta = \\frac{1}{\\tan\\theta}<\/annotation><\/semantics><\/math>csc\u03b8=sin\u03b81\u200bsec\u03b8=cos\u03b81\u200bcot\u03b8=tan\u03b81\u200b<\/p>\n\n\n\n

Quotient Identities<\/strong>tan<\/mi>\u2061<\/mo>\u03b8<\/mi>=<\/mo>sin<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow>cos<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow><\/mfrac><\/mspace>cot<\/mi>\u2061<\/mo>\u03b8<\/mi>=<\/mo>cos<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow>sin<\/mi>\u2061<\/mo>\u03b8<\/mi><\/mrow><\/mfrac><\/mrow>\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta} \\qquad \\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}<\/annotation><\/semantics><\/math>tan\u03b8=cos\u03b8sin\u03b8\u200bcot\u03b8=sin\u03b8cos\u03b8\u200b<\/p>\n\n\n\n

Common mistake: confusing cot with the reciprocal of tan vs the quotient of cos and sin \u2014 they are the same thing, but students who don’t see this connection make errors when substituting.<\/em><\/p>\n\n\n\n

\n\n\n\n

Math 30-1 Formula Sheet: Sum and Difference Formulas<\/h3>\n\n\n\n

sin<\/mi>\u2061<\/mo>(<\/mo>A<\/mi>\u00b1<\/mo>B<\/mi>)<\/mo>=<\/mo>sin<\/mi>\u2061<\/mo>A<\/mi>cos<\/mi>\u2061<\/mo>B<\/mi>\u00b1<\/mo>cos<\/mi>\u2061<\/mo>A<\/mi>sin<\/mi>\u2061<\/mo>B<\/mi><\/mrow>\\sin(A \\pm B) = \\sin A\\cos B \\pm \\cos A\\sin B<\/annotation><\/semantics><\/math>sin(A\u00b1B)=sinAcosB\u00b1cosAsinB cos<\/mi>\u2061<\/mo>(<\/mo>A<\/mi>\u00b1<\/mo>B<\/mi>)<\/mo>=<\/mo>cos<\/mi>\u2061<\/mo>A<\/mi>cos<\/mi>\u2061<\/mo>B<\/mi>\u2213<\/mo>sin<\/mi>\u2061<\/mo>A<\/mi>sin<\/mi>\u2061<\/mo>B<\/mi><\/mrow>\\cos(A \\pm B) = \\cos A\\cos B \\mp \\sin A\\sin B<\/annotation><\/semantics><\/math>cos(A\u00b1B)=cosAcosB\u2213sinAsinB tan<\/mi>\u2061<\/mo>(<\/mo>A<\/mi>\u00b1<\/mo>B<\/mi>)<\/mo>=<\/mo>tan<\/mi>\u2061<\/mo>A<\/mi>\u00b1<\/mo>tan<\/mi>\u2061<\/mo>B<\/mi><\/mrow>1<\/mn>\u2213<\/mo>tan<\/mi>\u2061<\/mo>A<\/mi>tan<\/mi>\u2061<\/mo>B<\/mi><\/mrow><\/mfrac><\/mrow>\\tan(A \\pm B) = \\frac{\\tan A \\pm \\tan B}{1 \\mp \\tan A\\tan B}<\/annotation><\/semantics><\/math>tan(A\u00b1B)=1\u2213tanAtanBtanA\u00b1tanB\u200b<\/p>\n\n\n\n

Note the sign reversal in the cosine formula (\u00b1 on the left becomes \u2213 on the right). This is the most commonly misremembered formula in this set \u2014 cos(A + B) expands with subtraction.<\/em><\/p>\n\n\n\n

\n\n\n\n

Double Angle Formulas<\/h3>\n\n\n\n

sin<\/mi>\u2061<\/mo>2<\/mn>A<\/mi>=<\/mo>2<\/mn>sin<\/mi>\u2061<\/mo>A<\/mi>cos<\/mi>\u2061<\/mo>A<\/mi><\/mrow>\\sin 2A = 2\\sin A\\cos A<\/annotation><\/semantics><\/math>sin2A=2sinAcosA cos<\/mi>\u2061<\/mo>2<\/mn>A<\/mi>=<\/mo>cos<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>A<\/mi>\u2212<\/mo>sin<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>A<\/mi>=<\/mo>2<\/mn>cos<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>A<\/mi>\u2212<\/mo>1<\/mn>=<\/mo>1<\/mn>\u2212<\/mo>2<\/mn>sin<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>A<\/mi><\/mrow>\\cos 2A = \\cos^2 A – \\sin^2 A = 2\\cos^2 A – 1 = 1 – 2\\sin^2 A<\/annotation><\/semantics><\/math>cos2A=cos2A\u2212sin2A=2cos2A\u22121=1\u22122sin2A tan<\/mi>\u2061<\/mo>2<\/mn>A<\/mi>=<\/mo>2<\/mn>tan<\/mi>\u2061<\/mo>A<\/mi><\/mrow>1<\/mn>\u2212<\/mo>tan<\/mi>\u2061<\/mo><\/mrow>2<\/mn><\/msup>A<\/mi><\/mrow><\/mfrac><\/mrow>\\tan 2A = \\frac{2\\tan A}{1 – \\tan^2 A}<\/annotation><\/semantics><\/math>tan2A=1\u2212tan2A2tanA\u200b<\/p>\n\n\n\n

The three forms of cos 2A are all equivalent \u2014 each is derived by substituting the Pythagorean identity into the cos\u00b2A \u2212 sin\u00b2A form. On the diploma exam, choosing the right form can simplify a proof significantly.<\/em><\/p>\n\n\n\n

\n\n\n\n

Radian and Degree Conversion<\/h3>\n\n\n\n

Radians<\/mtext>=<\/mo>Degrees<\/mtext>\u00d7<\/mo>\u03c0<\/mi>180<\/mn><\/mfrac><\/mrow>\\text{Radians} = \\text{Degrees} \\times \\frac{\\pi}{180}<\/annotation><\/semantics><\/math>Radians=Degrees\u00d7180\u03c0\u200b Degrees<\/mtext>=<\/mo>Radians<\/mtext>\u00d7<\/mo>180<\/mn>\u03c0<\/mi><\/mfrac><\/mrow>\\text{Degrees} = \\text{Radians} \\times \\frac{180}{\\pi}<\/annotation><\/semantics><\/math>Degrees=Radians\u00d7\u03c0180\u200b<\/p>\n\n\n\n

**Arc length formula:** a<\/mi>=<\/mo>r<\/mi>\u03b8<\/mi><\/mrow>a = r\\theta<\/annotation><\/semantics><\/math>a=r\u03b8<\/p>\n\n\n\n

*(where a = arc length, r = radius, \u03b8 = angle in radians)*<\/p>\n\n\n\n

Special angle values to know from memory:<\/strong><\/p>\n\n\n\n

Degrees<\/th>Radians<\/th>sin<\/th>cos<\/th>tan<\/th><\/tr><\/thead>
0\u00b0<\/td>0<\/td>0<\/td>1<\/td>0<\/td><\/tr>
30\u00b0<\/td>\u03c0\/6<\/td>1\/2<\/td>\u221a3\/2<\/td>1\/\u221a3<\/td><\/tr>
45\u00b0<\/td>\u03c0\/4<\/td>\u221a2\/2<\/td>\u221a2\/2<\/td>1<\/td><\/tr>
60\u00b0<\/td>\u03c0\/3<\/td>\u221a3\/2<\/td>1\/2<\/td>\u221a3<\/td><\/tr>
90\u00b0<\/td>\u03c0\/2<\/td>1<\/td>0<\/td>undefined<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

These must be committed to memory \u2014 the diploma exam will test them in radian form without a calculator permitted for Part A.<\/em><\/p>\n\n\n\n

\n\n\n\n

General Form of Sinusoidal Functions<\/h3>\n\n\n\n

y<\/mi>=<\/mo>a<\/mi>sin<\/mi>\u2061<\/mo>(<\/mo>b<\/mi>(<\/mo>x<\/mi>\u2212<\/mo>c<\/mi>)<\/mo>)<\/mo>+<\/mo>d<\/mi><\/mspace>y<\/mi>=<\/mo>a<\/mi>cos<\/mi>\u2061<\/mo>(<\/mo>b<\/mi>(<\/mo>x<\/mi>\u2212<\/mo>c<\/mi>)<\/mo>)<\/mo>+<\/mo>d<\/mi><\/mrow>y = a\\sin(b(x – c)) + d \\qquad y = a\\cos(b(x – c)) + d<\/annotation><\/semantics><\/math>y=asin(b(x\u2212c))+dy=acos(b(x\u2212c))+d<\/p>\n\n\n\n
Parameter<\/th>Effect<\/th><\/tr><\/thead>
|a|<\/td>Amplitude<\/td><\/tr>
b<\/td>Period = 2\u03c0\/|b| (for sin\/cos)<\/td><\/tr>
c<\/td>Phase shift (horizontal translation)<\/td><\/tr>
d<\/td>Vertical displacement (midline)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Common mistake: students confuse the period formula \u2014 period = 2\u03c0\/b, not 2\u03c0b. A larger b gives a shorter period.<\/em><\/p>\n\n\n\n

\n\n\n\n

Math 30-1 Formula Sheet: Functions Formulas<\/h2>\n\n\n\n

Transformations: General Form<\/h3>\n\n\n\n

For any function f(x): y<\/mi>=<\/mo>a<\/mi>f<\/mi>(<\/mo>b<\/mi>(<\/mo>x<\/mi>\u2212<\/mo>c<\/mi>)<\/mo>)<\/mo>+<\/mo>d<\/mi><\/mrow>y = af(b(x – c)) + d<\/annotation><\/semantics><\/math>y=af(b(x\u2212c))+d<\/p>\n\n\n\n
Parameter<\/th>Transformation<\/th><\/tr><\/thead>
a > 1<\/td>Vertical stretch by factor a<\/td><\/tr>
0 < a < 1<\/td>Vertical compression by factor a<\/td><\/tr>
a < 0<\/td>Reflection over x-axis<\/td><\/tr>
b > 1<\/td>Horizontal compression by factor 1\/b<\/td><\/tr>
0 < b < 1<\/td>Horizontal stretch by factor 1\/b<\/td><\/tr>
b < 0<\/td>Reflection over y-axis<\/td><\/tr>
c > 0<\/td>Shift right c units<\/td><\/tr>
c < 0<\/td>Shift left |c| units<\/td><\/tr>
d > 0<\/td>Shift up d units<\/td><\/tr>
d < 0<\/td>Shift down |d| units<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Common mistake: students reverse the direction of horizontal transformations. The shift is c units in the direction of the sign inside the bracket \u2014 so f(x \u2212 3) shifts right 3, not left.<\/em><\/p>\n\n\n\n

\n\n\n\n

Inverse Functions<\/h3>\n\n\n\n

If f and f\u207b\u00b9 are inverse functions: f<\/mi>(<\/mo>f<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>x<\/mi>)<\/mo>)<\/mo>=<\/mo>x<\/mi><\/mspace>f<\/mi>\u2212<\/mo>1<\/mn><\/mrow><\/msup>(<\/mo>f<\/mi>(<\/mo>x<\/mi>)<\/mo>)<\/mo>=<\/mo>x<\/mi><\/mrow>f(f^{-1}(x)) = x \\qquad f^{-1}(f(x)) = x<\/annotation><\/semantics><\/math>f(f\u22121(x))=xf\u22121(f(x))=x<\/p>\n\n\n\n

To find an inverse algebraically: swap x and y, then solve for y.<\/p>\n\n\n\n

The graph of f\u207b\u00b9 is the reflection of f over the line y = x.<\/p>\n\n\n\n

Domain restriction for inverses:<\/strong> An inverse is only a function if the original function is one-to-one (passes the horizontal line test). When restricting domain, the most common convention is to restrict to the interval where the function is increasing.<\/p>\n\n\n\n

\n\n\n\n

Absolute Value<\/h3>\n\n\n\n

\u2223<\/mi>x<\/mi>\u2223<\/mi>=<\/mo>{<\/mo>x<\/mi><\/mstyle><\/mtd>if <\/mtext>x<\/mi>\u2265<\/mo>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr>\u2212<\/mo>x<\/mi><\/mrow><\/mstyle><\/mtd>if <\/mtext>x<\/mi><<\/mo>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow>|x| = \\begin{cases} x & \\text{if } x \\geq 0 \\\\ -x & \\text{if } x < 0 \\end{cases}<\/annotation><\/semantics><\/math>\u2223x\u2223={x\u2212x\u200bif x\u22650if x<0\u200b<\/p>\n\n\n\n

For solving |f(x)| = a (where a > 0): f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>a<\/mi><\/mspace>or<\/mtext><\/mspace>f<\/mi>(<\/mo>x<\/mi>)<\/mo>=<\/mo>\u2212<\/mo>a<\/mi><\/mrow>f(x) = a \\quad \\text{or} \\quad f(x) = -a<\/annotation><\/semantics><\/math>f(x)=aorf(x)=\u2212a<\/p>\n\n\n\n

For solving |f(x)| < a: \u2212<\/mo>a<\/mi><<\/mo>f<\/mi>(<\/mo>x<\/mi>)<\/mo><<\/mo>a<\/mi><\/mrow>-a < f(x) < a<\/annotation><\/semantics><\/math>\u2212a<f(x)<a<\/p>\n\n\n\n

For solving |f(x)| > a: f<\/mi>(<\/mo>x<\/mi>)<\/mo><<\/mo>\u2212<\/mo>a<\/mi><\/mspace>or<\/mtext><\/mspace>f<\/mi>(<\/mo>x<\/mi>)<\/mo>><\/mo>a<\/mi><\/mrow>f(x) < -a \\quad \\text{or} \\quad f(x) > a<\/annotation><\/semantics><\/math>f(x)<\u2212aorf(x)>a<\/p>\n\n\n\n
\n\n\n\n

Rational Functions: Asymptotes<\/h3>\n\n\n\n

For a rational function f(x) = P(x)\/Q(x):<\/p>\n\n\n\n

Vertical asymptotes:<\/strong> occur where Q(x) = 0 and P(x) \u2260 0 (if both equal zero, there may be a hole instead)<\/p>\n\n\n\n

Horizontal asymptotes:<\/strong><\/p>\n\n\n\n