Trigonometric Identities cheat sheet - grade 10-12

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Math Grade 10-12

Trigonometric Identities Cheat Sheet

A printable reference covering reciprocal, quotient, Pythagorean, sum, difference, double-angle, half-angle, and cofunction identities for grades 10-12.

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Trigonometric identities are equations that are true for all allowed angle values. Students need them to simplify expressions, solve equations, verify relationships, and connect graphs with the unit circle. This cheat sheet organizes the main identity families used in grades 10-12 so students can choose the right tool quickly. The most important identities come from the unit circle and the relationship between sine, cosine, and tangent. Core ideas include rewriting functions with reciprocal and quotient identities, using sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, and applying sum and difference formulas. Double-angle and half-angle identities help transform expressions involving 2θ2\theta or θ2\frac{\theta}{2} into easier forms.

Key Facts

  • The reciprocal identities are csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, and cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}.
  • The quotient identities are tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
  • The main Pythagorean identity is sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Dividing sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 by cos⁡2θ\cos^2 \theta gives 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta.
  • Dividing sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 by sin⁡2θ\sin^2 \theta gives 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta.
  • The sum and difference identities are sin⁡(a±b)=sin⁡acos⁡b±cos⁡asin⁡b\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b and cos⁡(a±b)=cos⁡acos⁡b∓sin⁡asin⁡b\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b.
  • The double-angle identities include sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta and cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2 \theta - \sin^2 \theta.
  • The half-angle identities are sin⁡2(θ2)=1−cos⁡θ2\sin^2 \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{2} and cos⁡2(θ2)=1+cos⁡θ2\cos^2 \left(\frac{\theta}{2}\right) = \frac{1 + \cos \theta}{2}.

Vocabulary

Trigonometric identity
A trigonometric identity is an equation involving trigonometric functions that is true for every angle where both sides are defined.
Reciprocal identity
A reciprocal identity rewrites a trigonometric function as the reciprocal of another function, such as sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}.
Quotient identity
A quotient identity expresses tangent or cotangent as a ratio, such as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.
Pythagorean identity
A Pythagorean identity connects squared trigonometric functions, with the main one being sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
Cofunction identity
A cofunction identity relates complementary angles, such as sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos(90^\circ - \theta).
Double-angle identity
A double-angle identity rewrites a function of 2θ2\theta using functions of θ\theta, such as sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta.

Common Mistakes to Avoid

  • Writing sin⁡2θ\sin^2 \theta as sin⁡(2θ)\sin(2\theta) is wrong because sin⁡2θ\sin^2 \theta means (sin⁡θ)2(\sin \theta)^2, not the sine of a doubled angle.
  • Changing the sign incorrectly in cos⁡(a±b)\cos(a \pm b) is wrong because cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin b and cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos(a - b) = \cos a \cos b + \sin a \sin b.
  • Canceling across sums such as sin⁡θ+1sin⁡θ\frac{\sin \theta + 1}{\sin \theta} is wrong because terms in a sum cannot be canceled separately unless they are common factors.
  • Using tan⁡θ=cos⁡θsin⁡θ\tan \theta = \frac{\cos \theta}{\sin \theta} is wrong because tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
  • Forgetting domain restrictions is wrong because identities like tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} are only valid where cos⁡θ≠0\cos \theta \ne 0.

Practice Questions

  1. 1 Simplify 1−cos⁡2θsin⁡θ\frac{1 - \cos^2 \theta}{\sin \theta} using a Pythagorean identity.
  2. 2 Find the exact value of sin⁡(75∘)\sin(75^\circ) using sin⁡(a+b)\sin(a + b).
  3. 3 Rewrite cos⁡(2θ)\cos(2\theta) in two different equivalent forms using Pythagorean identities.
  4. 4 Explain why proving an identity usually works better by transforming one side into the other instead of substituting one angle value.