{"id":5347,"date":"2025-12-16T09:13:17","date_gmt":"2025-12-16T15:13:17","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=5347"},"modified":"2025-12-16T09:19:38","modified_gmt":"2025-12-16T15:19:38","slug":"trigonometry-cheat-sheet","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/math-resources\/trigonometry-cheat-sheet\/","title":{"rendered":"Trigonometry Cheat Sheet"},"content":{"rendered":"\n

Definition of the Trig Functions<\/h2>\n\n\n\n
\n
\n

Right triangle definition<\/strong>
For this definition we assume that<\/p>\n\n\n\n$$\n0 < \\theta < \\frac{\\pi}{2} \\quad \\text{or} \\quad 0^\\circ < \\theta < 90^\\circ\n$$\n\n\n

\n
\"Right<\/figure>\n<\/div>\n\n\n
\n
\n$$\n\\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}\n\\qquad\n$$\n$$\n\\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}\n\\qquad\n$$\n$$\n\\tan \\theta = \\frac{\\text{opposite}}{\\text{adjacent}}\n\\qquad\n$$\n<\/div>\n\n\n\n
\n$$\n\\csc \\theta = \\frac{\\text{hypotenuse}}{\\text{opposite}}\n$$\n\n$$\n\\sec \\theta = \\frac{\\text{hypotenuse}}{\\text{adjacent}}\n$$\n\n$$\n\\cot \\theta = \\frac{\\text{adjacent}}{\\text{opposite}}\n$$\n<\/div>\n<\/div>\n<\/div>\n\n\n\n
\n

Unit circle definition
<\/strong>For this definition \u03b8<\/em> is any angle.<\/p>\n\n\n

\n
\"Unit<\/figure>\n<\/div>\n\n\n$$\n\\sin \\theta = \\frac{y}{1} = y\n\\qquad\n\\csc \\theta = \\frac{1}{y}\n$$\n\n$$\n\\cos \\theta = \\frac{x}{1} = x\n\\qquad\n\\sec \\theta = \\frac{1}{x}\n$$\n\n$$\n\\tan \\theta = \\frac{y}{x}\n\\qquad\n \n\\cot \\theta = \\frac{x}{y}\n$$\n<\/div>\n<\/div>\n\n\n\n
<\/div>\n\n\n\n

Facts and Properties<\/h2>\n\n\n\n
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Domain<\/strong>
The domain is all the values of \u03b8<\/em> that
can be plugged into the function.<\/p>\n\n\n\n$$\n\\sin \\theta,\\quad \\theta \\text{ can be any angle}\n$$\n\n$$\n\\cos \\theta,\\quad \\theta \\text{ can be any angle}\n$$\n\n$$\n\\tan \\theta,\\quad \\theta \\ne \\left(n + \\frac{1}{2}\\right)\\pi,\\; n = 0, \\pm 1, \\pm 2, \\ldots\n$$\n\n$$\n\\csc \\theta,\\quad \\theta \\ne n\\pi,\\; n = 0, \\pm 1, \\pm 2, \\ldots\n$$\n\n$$\n\\sec \\theta,\\quad \\theta \\ne \\left(n + \\frac{1}{2}\\right)\\pi,\\; n = 0, \\pm 1, \\pm 2, \\ldots\n$$\n\n$$\n\\cot \\theta,\\quad \\theta \\ne n\\pi,\\; n = 0, \\pm 1, \\pm 2, \\ldots\n$$\n\n\n\n

Range<\/strong>
The range is all possible values to get
out of the function.<\/p>\n\n\n\n$$\n-1 \\le \\sin \\theta \\le 1\n\\qquad\n\\csc \\theta \\ge 1 \\;\\text{and}\\; \\csc \\theta \\le -1\n$$\n\n$$\n-1 \\le \\cos \\theta \\le 1\n\\qquad\n\\sec \\theta \\ge 1 \\;\\text{and}\\; \\sec \\theta \\le -1\n$$\n\n$$\n-\\infty < \\tan \\theta < \\infty\n\\qquad\n-\\infty < \\cot \\theta < \\infty\n$$\n<\/div>\n\n\n\n

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Period<\/strong>
The period of a function is the number, T<\/em>, such that f<\/em>(\u03b8<\/em> + T<\/em>) = f<\/em>(\u03b8<\/em>). So, if \u03c9<\/em> is a fixed number and \u03b8<\/em> is any angle, we have the following periods.<\/p>\n\n\n\n$$\n\\sin(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{2\\pi}{\\omega}\n$$\n\n$$\n\\cos(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{2\\pi}{\\omega}\n$$\n\n$$\n\\tan(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{\\pi}{\\omega}\n$$\n\n$$\n\\csc(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{2\\pi}{\\omega}\n$$\n\n$$\n\\sec(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{2\\pi}{\\omega}\n$$\n\n$$\n\\cot(\\omega \\theta) \\;\\rightarrow\\; T = \\frac{\\pi}{\\omega}\n$$\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n

Formulas and Identities<\/h2>\n\n\n\n
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Tangent and Cotangent Identities<\/strong><\/p>\n\n\n\n$$\n\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}\n\\qquad\n\\cot \\theta = \\frac{\\cos \\theta}{\\sin \\theta}\n$$\n\n\n\n

Reciprocal Identities<\/strong><\/p>\n\n\n\n$$\n\\sec \\theta = \\frac{1}{\\cos \\theta}\n\\qquad\n\\cos \\theta = \\frac{1}{\\sec \\theta}\n$$\n\n$$\n\\cot \\theta = \\frac{1}{\\tan \\theta}\n\\qquad\n\\tan \\theta = \\frac{1}{\\cot \\theta}\n$$\n\n\n\n

Pythagorean Identities<\/strong><\/p>\n\n\n\n$$\n\\sin^2 \\theta + \\cos^2 \\theta = 1\n$$\n\n$$\n\\tan^2 \\theta + 1 = \\sec^2 \\theta\n$$\n\n$$\n1 + \\cot^2 \\theta = \\csc^2 \\theta\n$$\n\n\n\n\n

Even\/Odd Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin(-\\theta) = -\\sin \\theta\n\\qquad\n\\csc(-\\theta) = -\\csc \\theta\n$$\n\n$$\n\\cos(-\\theta) = \\cos \\theta\n\\qquad\n\\sec(-\\theta) = \\sec \\theta\n$$\n\n$$\n\\tan(-\\theta) = -\\tan \\theta\n\\qquad\n\\cot(-\\theta) = -\\cot \\theta\n$$\n\n\n\n\n

Periodic Formulas<\/strong>
If n<\/em> is an integer.<\/p>\n\n\n\n$$\n\\sin(\\theta + 2\\pi n) = \\sin \\theta\n\\qquad\n\\csc(\\theta + 2\\pi n) = \\csc \\theta\n$$\n\n$$\n\\cos(\\theta + 2\\pi n) = \\cos \\theta\n\\qquad\n\\sec(\\theta + 2\\pi n) = \\sec \\theta\n$$\n\n$$\n\\tan(\\theta + \\pi n) = \\tan \\theta\n\\qquad\n\\cot(\\theta + \\pi n) = \\cot \\theta\n$$\n\n\n\n\n

Double Angle Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin(2\\theta) = 2 \\sin \\theta \\cos \\theta\n$$\n\n$$\n\\cos(2\\theta) = \\cos^2 \\theta – \\sin^2 \\theta\n$$\n\n$$\n\\cos(2\\theta) = 2 \\cos^2 \\theta – 1\n$$\n\n$$\n\\cos(2\\theta) = 1 – 2 \\sin^2 \\theta\n$$\n\n$$\n\\tan(2\\theta) = \\frac{2 \\tan \\theta}{1 – \\tan^2 \\theta}\n$$\n\n\n\n\n

Degrees to Radians Formulas<\/strong>
If x<\/em> is an angle in degrees and t<\/em> is an
angle in radians then<\/p>\n\n\n\n$$\n\\frac{\\pi}{180} = \\frac{t}{x}\n\\;\\Rightarrow\\;\nt = \\frac{\\pi x}{180}\n\\;\\text{and}\\;\nx = \\frac{180t}{\\pi}\n$$\n\n<\/div>\n\n\n\n

\n
\n
\n

Half Angle Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin \\frac{\\theta}{2}\n= \\pm \\sqrt{\\frac{1 – \\cos \\theta}{2}}\n$$\n\n$$\n\\cos \\frac{\\theta}{2}\n= \\pm \\sqrt{\\frac{1 + \\cos \\theta}{2}}\n$$\n\n\n\n$$\n\\tan \\frac{\\theta}{2}\n= \\pm \\sqrt{\\frac{1 – \\cos \\theta}{1 + \\cos \\theta}}\n$$\n<\/div>\n\n\n\n

\n

(alternate form)<\/strong><\/p>\n\n\n\n$$\n\\sin^2 \\theta\n= \\frac{1}{2}\\bigl(1 – \\cos(2\\theta)\\bigr)\n$$\n\n$$\n\\cos^2 \\theta\n= \\frac{1}{2}\\bigl(1 + \\cos(2\\theta)\\bigr)\n$$\n\n$$\n\\tan^2 \\theta\n= \\frac{1 – \\cos(2\\theta)}{1 + \\cos(2\\theta)}\n$$\n<\/div>\n<\/div>\n\n\n\n

Sum and Difference Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin(\\alpha \\pm \\beta)\n= \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta\n$$\n\n$$\n\\cos(\\alpha \\pm \\beta)\n= \\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta\n$$\n\n$$\n\\tan(\\alpha \\pm \\beta)\n= \\frac{\\tan \\alpha \\pm \\tan \\beta}{1 \\mp \\tan \\alpha \\tan \\beta}\n$$\n\n\n\n

Product to Sum Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin \\alpha \\sin \\beta\n= \\frac{1}{2}\\bigl[\\cos(\\alpha – \\beta) – \\cos(\\alpha + \\beta)\\bigr]\n$$\n\n$$\n\\cos \\alpha \\cos \\beta\n= \\frac{1}{2}\\bigl[\\cos(\\alpha – \\beta) + \\cos(\\alpha + \\beta)\\bigr]\n$$\n\n$$\n\\sin \\alpha \\cos \\beta\n= \\frac{1}{2}\\bigl[\\sin(\\alpha + \\beta) + \\sin(\\alpha – \\beta)\\bigr]\n$$\n\n$$\n\\cos \\alpha \\sin \\beta\n= \\frac{1}{2}\\bigl[\\sin(\\alpha + \\beta) – \\sin(\\alpha – \\beta)\\bigr]\n$$\n\n\n\n

Sum to Product Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin \\alpha + \\sin \\beta\n= 2 \\sin\\!\\left(\\frac{\\alpha + \\beta}{2}\\right)\n \\cos\\!\\left(\\frac{\\alpha – \\beta}{2}\\right)\n$$\n\n$$\n\\sin \\alpha – \\sin \\beta\n= 2 \\cos\\!\\left(\\frac{\\alpha + \\beta}{2}\\right)\n \\sin\\!\\left(\\frac{\\alpha – \\beta}{2}\\right)\n$$\n\n$$\n\\cos \\alpha + \\cos \\beta\n= 2 \\cos\\!\\left(\\frac{\\alpha + \\beta}{2}\\right)\n \\cos\\!\\left(\\frac{\\alpha – \\beta}{2}\\right)\n$$\n\n$$\n\\cos \\alpha – \\cos \\beta\n= -2 \\sin\\!\\left(\\frac{\\alpha + \\beta}{2}\\right)\n \\sin\\!\\left(\\frac{\\alpha – \\beta}{2}\\right)\n$$\n\n\n\n

Cofunction Formulas<\/strong><\/p>\n\n\n\n$$\n\\sin\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\cos \\theta\n\\qquad\n\\cos\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\sin \\theta\n$$\n\n$$\n\\csc\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\sec \\theta\n\\qquad\n\\sec\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\csc \\theta\n$$\n\n$$\n\\tan\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\cot \\theta\n\\qquad\n\\cot\\!\\left(\\frac{\\pi}{2} – \\theta\\right) = \\tan \\theta\n$$\n\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n

Unit Circle<\/h2>\n\n\n
\n
\"Unit<\/figure>\n<\/div>\n\n\n

For any ordered pair on the unit circle ( x<\/em>, y<\/em> ) : cos \u03b8<\/em> = x<\/em> and sin \u03b8<\/em> = y<\/em><\/p>\n\n\n\n

Example<\/strong><\/p>\n\n\n\n$$\n\\cos\\!\\left(\\frac{5\\pi}{3}\\right) = \\frac{1}{2}\n\\qquad\n\\sin\\!\\left(\\frac{5\\pi}{3}\\right) = -\\frac{\\sqrt{3}}{2}\n$$\n\n\n\n

<\/div>\n\n\n\n

Inverse Trig Functions<\/h2>\n\n\n\n
\n
\n

Definition<\/strong>
y<\/em> = sin\u22121<\/sup> x<\/em> is equivalent to x<\/em> = sin y<\/em>
y<\/em> = cos\u22121<\/sup> x<\/em> is equivalent to x<\/em> = cos y<\/em>
y<\/em> = tan\u22121<\/sup> x<\/em> is equivalent to x<\/em> = tan y<\/em><\/p>\n\n\n\n

Domain and Range<\/strong><\/p>\n\n\n\n$$\n\\begin{array}{c c c}\n\\textbf{Function} & \\textbf{Domain} & \\textbf{Range} \\\\[6pt]\ny = \\sin^{-1} x & -1 \\le x \\le 1 & -\\dfrac{\\pi}{2} \\le y \\le \\dfrac{\\pi}{2} \\\\[6pt]\ny = \\cos^{-1} x & -1 \\le x \\le 1 & 0 \\le y \\le \\pi \\\\[6pt]\ny = \\tan^{-1} x & -\\infty < x < \\infty & -\\dfrac{\\pi}{2} < y < \\dfrac{\\pi}{2}\n\\end{array}\n$$\n<\/div>\n\n\n\n

\n

Inverse Properties<\/strong><\/p>\n\n\n\n$$\n\\cos\\!\\bigl(\\cos^{-1}(x)\\bigr) = x\n\\qquad\n\\cos^{-1}\\!\\bigl(\\cos(\\theta)\\bigr) = \\theta\n$$\n\n$$\n\\sin\\!\\bigl(\\sin^{-1}(x)\\bigr) = x\n\\qquad\n\\sin^{-1}\\!\\bigl(\\sin(\\theta)\\bigr) = \\theta\n$$\n\n$$\n\\tan\\!\\bigl(\\tan^{-1}(x)\\bigr) = x\n\\qquad\n\\tan^{-1}\\!\\bigl(\\tan(\\theta)\\bigr) = \\theta\n$$\n\n\n\n

Alternate Notation<\/strong><\/p>\n\n\n\n$$\n\\sin^{-1} x = \\arcsin x\n$$\n\n$$\n\\cos^{-1} x = \\arccos x\n$$\n\n$$\n\\tan^{-1} x = \\arctan x\n$$\n\n<\/div>\n<\/div>\n\n\n\n

<\/div>\n\n\n\n

Law of Sines, Cosines and Tangents<\/h2>\n\n\n
\n
\"Scalene<\/figure>\n<\/div>\n\n\n
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Law of Sines<\/strong><\/p>\n\n\n\n$$\n\\frac{\\sin \\alpha}{a}\n= \\frac{\\sin \\beta}{b}\n= \\frac{\\sin \\gamma}{c}\n$$\n\n\n\n

Law of Cosines<\/strong><\/p>\n\n\n\n$$\na^2 = b^2 + c^2 – 2bc \\cos \\alpha\n$$\n\n$$\nb^2 = a^2 + c^2 – 2ac \\cos \\beta\n$$\n\n$$\nc^2 = a^2 + b^2 – 2ab \\cos \\gamma\n$$\n\n\n\n

Mollweide\u2019s Formula<\/strong><\/p>\n\n\n\n$$\n\\frac{a + b}{c}\n= \\frac{\\cos\\!\\left(\\tfrac{1}{2}(\\alpha – \\beta)\\right)}\n {\\sin\\!\\left(\\tfrac{1}{2}\\gamma\\right)}\n$$\n<\/div>\n\n\n\n

\n

Law of Tangents<\/strong><\/p>\n\n\n\n$$\n\\frac{a – b}{a + b}\n= \\frac{\\tan\\!\\left(\\tfrac{1}{2}(\\alpha – \\beta)\\right)}\n {\\tan\\!\\left(\\tfrac{1}{2}(\\alpha + \\beta)\\right)}\n$$\n\n$$\n\\frac{b – c}{b + c}\n= \\frac{\\tan\\!\\left(\\tfrac{1}{2}(\\beta – \\gamma)\\right)}\n {\\tan\\!\\left(\\tfrac{1}{2}(\\beta + \\gamma)\\right)}\n$$\n\n$$\n\\frac{a – c}{a + c}\n= \\frac{\\tan\\!\\left(\\tfrac{1}{2}(\\alpha – \\gamma)\\right)}\n {\\tan\\!\\left(\\tfrac{1}{2}(\\alpha + \\gamma)\\right)}\n$$\n<\/div>\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Definition of the Trig Functions Right triangle definitionFor this definition we assume that $$ 0 < \\theta < \\frac{\\pi}{2} \\quad \\text{or} \\quad 0^\\circ < \\theta < 90^\\circ $$ $$ \\sin \\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}} \\qquad $$ $$ \\cos \\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}} \\qquad $$ $$ \\tan \\theta = \\frac{\\text{opposite}}{\\text{adjacent}} \\qquad $$ $$ \\csc \\theta = \\frac{\\text{hypotenuse}}{\\text{opposite}} $$ […]\n<\/p>\n","protected":false},"author":98,"featured_media":0,"parent":3836,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template\/template-full-page.php","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-5347","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/5347","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/98"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=5347"}],"version-history":[{"count":48,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/5347\/revisions"}],"predecessor-version":[{"id":5411,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/5347\/revisions\/5411"}],"up":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/3836"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=5347"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=5347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}