Trigonometry

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high importance~3 Q in Tier 119 formulas⚡ 15 shortcuts5 subtopics

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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Ratios and standard values

Primary ratios

\sin\theta=\frac{o}{h},\quad \cos\theta=\frac{a}{h},\quad \tan\theta=\frac{o}{a}

o = opposite, a = adjacent, h = hypotenuse.

Reciprocals

\text{cosec}=\frac{1}{\sin},\quad \sec=\frac{1}{\cos},\quad \cot=\frac{1}{\tan}

Flip the fraction.

Standard values

\sin\theta=\frac{\sqrt{k}}{2},\ k=0,1,2,3,4

For 0, 30, 45, 60, 90 degrees; cos runs the row backwards.

One ratio to all

\sin\theta=\frac{3}{5}\Rightarrow\cos=\frac45,\ \tan=\frac34

Draw the 3-4-5 triangle and read every ratio off it.

Fundamental identities

Pythagorean identities

\sin^2+\cos^2=1,\quad 1+\tan^2=\sec^2,\quad 1+\cot^2=\cosec^2

Three engines from one identity.

Conjugate pairs

(\sec+\tan)(\sec-\tan)=1,\quad (\cosec+\cot)(\cosec-\cot)=1

Sum and difference are reciprocals.

Squares of sums

(a+b)^2+(a-b)^2=2(a^2+b^2)

Cross terms cancel in pairs.

Reciprocal products

\sin\cdot\cosec=\cos\cdot\sec=\tan\cdot\cot=1

The constant that kills cross terms.

Complementary angles

Complementary swaps

\sin(90^\circ-\theta)=\cos\theta,\ \tan(90^\circ-\theta)=\cot\theta,\ \sec(90^\circ-\theta)=\cosec\theta

Drop the co- or add it.

Right triangle angles

A+B=90^\circ\Rightarrow\sin A=\cos B

The two acute angles are partners.

Pairing to one

\tan\theta\cdot\tan(90^\circ-\theta)=1

Complementary tangents multiply to 1.

Value-putting and given-ratio questions

Square of sin+cos

(\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta

The bridge from a sum to a product.

Divide by cosine

\frac{a\sin+b\cos}{c\sin+d\cos}=\frac{a\tan+b}{c\tan+d}

After dividing every term by cosine.

Cot fraction to triangle

\cot\theta=\frac{21}{20}\Rightarrow\text{hyp}=29

Two sides given, Pythagoras gives the third.

Reciprocal pair sum

x+\frac{1}{x}\ \text{from}\ x\cdot\frac{1}{x}=1

Conjugates of cosec plus cot.

Maximum and minimum values

Amplitude of a sin + b cos

\max=\sqrt{a^2+b^2},\quad \min=-\sqrt{a^2+b^2}

General angles; on 0 to 90 check the endpoints too.

sin times cos

\sin\theta\cos\theta=\frac{\sin2\theta}{2}\le\frac12

Peak at 45 degrees.

AM-GM floor

x+\frac{1}{x}\ge2

For positive x; equality when x = 1.

Weighted squares

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

Rewrite as one constant plus one square.