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Trigonometry Formula Sheet

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Formula Sheet • Right & Non-Right Triangles

Trigonometry Formula Sheet

All essential trig formulas in one place — SOHCAHTOA for right-angled triangles, the Sine Rule and Cosine Rule for non-right triangles, triangle area, exact values, and core identities.

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SOHCAHTOA — Right-Angled Triangles

Sine Ratio (SOH)
sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}

Opposite = side across from angle θ • Hypotenuse = longest side across from 90°.

Worked Example
Opp=6,  Hyp=10  ⟹  sin⁡(θ)=610=0.6\displaystyle \text{Opp}=6,\; \text{Hyp}=10 \implies \sin(\theta) = \frac{6}{10} = 0.6
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Cosine Ratio (CAH)
cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

Adjacent = side next to angle θ (between θ and the right angle).

Worked Example
Adj=8,  Hyp=10  ⟹  cos⁡(θ)=810=0.8\displaystyle \text{Adj}=8,\; \text{Hyp}=10 \implies \cos(\theta) = \frac{8}{10} = 0.8
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Tangent Ratio (TOA)
tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

Relates the two perpendicular legs directly without needing the hypotenuse.

Worked Example
Opp=6,  Adj=8  ⟹  tan⁡(θ)=68=0.75\displaystyle \text{Opp}=6,\; \text{Adj}=8 \implies \tan(\theta) = \frac{6}{8} = 0.75
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Find Angle • Inverse Sine
θ=arcsin⁡ ⁣(OppHyp)\theta = \arcsin\!\left(\frac{\text{Opp}}{\text{Hyp}}\right)

Use when the opposite side and hypotenuse lengths are known.

Worked Example
θ=arcsin⁡(0.5)=30∘\displaystyle \theta = \arcsin(0.5) = 30^\circ
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Find Angle • Inverse Cosine
θ=arccos⁡ ⁣(AdjHyp)\theta = \arccos\!\left(\frac{\text{Adj}}{\text{Hyp}}\right)

Use when the adjacent side and hypotenuse lengths are known.

Worked Example
θ=arccos⁡(0.5)=60∘\displaystyle \theta = \arccos(0.5) = 60^\circ
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Find Angle • Inverse Tangent
θ=arctan⁡ ⁣(OppAdj)\theta = \arctan\!\left(\frac{\text{Opp}}{\text{Adj}}\right)

Use when both perpendicular legs are known.

Worked Example
θ=arctan⁡(1.0)=45∘\displaystyle \theta = \arctan(1.0) = 45^\circ
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Sine & Cosine Rules — Any Triangle

Sine Rule — Find a Side
asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Used when you know 2 angles and 1 side (AAS / ASA), or an opposite pair.

Worked Example
a=8×sin⁡(40∘)sin⁡(60∘)≈5.94\displaystyle a = \frac{8 \times \sin(40^\circ)}{\sin(60^\circ)} \approx 5.94
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Sine Rule — Find an Angle
sin⁡(A)a=sin⁡(B)b=sin⁡(C)c\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}

Inverted form makes rearranging simpler when solving for an unknown angle.

Worked Example
sin⁡(A)=5×sin⁡(30∘)7≈0.357  ⟹  A≈20.9∘\displaystyle \sin(A) = \frac{5 \times \sin(30^\circ)}{7} \approx 0.357 \implies A \approx 20.9^\circ
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Cosine Rule — Find a Side (SAS)
a2=b2+c2−2bccos⁡(A)a^2 = b^2 + c^2 - 2bc\cos(A)

Use when 2 sides and the included angle are known (SAS).

Worked Example
a2=72+102−2(7)(10)cos⁡(50∘)≈59.0  ⟹  a≈7.68\displaystyle a^2 = 7^2 + 10^2 - 2(7)(10)\cos(50^\circ) \approx 59.0 \implies a \approx 7.68
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Cosine Rule — Find an Angle (SSS)
cos⁡(A)=b2+c2−a22bc\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}

Use when all 3 sides are known (SSS) to find any interior angle.

Worked Example
cos⁡(A)=52+62−722(5)(6)=1260=0.2  ⟹  A≈78.5∘\displaystyle \cos(A) = \frac{5^2 + 6^2 - 7^2}{2(5)(6)} = \frac{12}{60} = 0.2 \implies A \approx 78.5^\circ
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Area of Any Triangle (SAS)
Area=12absin⁡(C)\text{Area} = \frac{1}{2}ab\sin(C)

Calculates the exact area from any two sides and the included angle C.

Worked Example
Area=12(7)(10)sin⁡(50∘)≈26.81 cm2\displaystyle \text{Area} = \frac{1}{2}(7)(10)\sin(50^\circ) \approx 26.81\text{ cm}^2
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Ambiguous Case (SSA)
B1=arcsin⁡(x),B2=180∘−B1B_1 = \arcsin(x), \quad B_2 = 180^\circ - B_1

When given two sides and a non-included acute angle, two valid triangles may exist if B2 + A < 180°.

Worked Example
B1=40∘  ⟹  B2=180∘−40∘=140∘\displaystyle B_1 = 40^\circ \implies B_2 = 180^\circ - 40^\circ = 140^\circ
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Identities & Exact Values

Tangent Quotient Identity
tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Holds true for all angles where cos(θ) ≠ 0.

Worked Example
θ=45∘:2/22/2=1=tan⁡(45∘)\displaystyle \theta=45^\circ: \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1 = \tan(45^\circ)
Pythagorean Identity
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1

Fundamental unit circle identity connecting sine and cosine for any angle.

Worked Example
sin⁡(θ)=0.6  ⟹  cos⁡(θ)=1−0.62=0.8\displaystyle \sin(\theta)=0.6 \implies \cos(\theta) = \sqrt{1 - 0.6^2} = 0.8
Elevation & Depression
Angle of Elevation=Angle of Depression\text{Angle of Elevation} = \text{Angle of Depression}

Measured from the horizontal sightline; equal due to alternate interior angles.

Worked Example
Horizontal distance d=htan⁡(Elevation)\displaystyle \text{Horizontal distance } d = \frac{h}{\tan(\text{Elevation})}
Exact Values: 30° (π/6 rad)
sin⁡30∘=12,  cos⁡30∘=32,  tan⁡30∘=13\sin 30^\circ = \frac{1}{2}, \; \cos 30^\circ = \frac{\sqrt{3}}{2}, \; \tan 30^\circ = \frac{1}{\sqrt{3}}

Derived from an equilateral triangle split in half with hypotenuse 2.

Worked Example
10×sin⁡(30∘)=10×12=5\displaystyle 10 \times \sin(30^\circ) = 10 \times \frac{1}{2} = 5
Exact Values: 45° (π/4 rad)
sin⁡45∘=22,  cos⁡45∘=22,  tan⁡45∘=1\sin 45^\circ = \frac{\sqrt{2}}{2}, \; \cos 45^\circ = \frac{\sqrt{2}}{2}, \; \tan 45^\circ = 1

Derived from an isosceles right triangle with legs of length 1.

Worked Example
42×cos⁡(45∘)=42×22=4\displaystyle 4\sqrt{2} \times \cos(45^\circ) = 4\sqrt{2} \times \frac{\sqrt{2}}{2} = 4
Exact Values: 60° (π/3 rad)
sin⁡60∘=32,  cos⁡60∘=12,  tan⁡60∘=3\sin 60^\circ = \frac{\sqrt{3}}{2}, \; \cos 60^\circ = \frac{1}{2}, \; \tan 60^\circ = \sqrt{3}

Complementary to 30°: sin(60°) = cos(30°) and cos(60°) = sin(30°).

Worked Example
6×tan⁡(60∘)=63≈10.39\displaystyle 6 \times \tan(60^\circ) = 6\sqrt{3} \approx 10.39
Angle θ (Deg) Angle θ (Rad) sin(θ) cos(θ) tan(θ)
0∘\displaystyle 0^\circ0\displaystyle 00\displaystyle 01\displaystyle 10\displaystyle 0
30∘\displaystyle 30^\circπ6\displaystyle \frac{\pi}{6}12\displaystyle \frac{1}{2}32\displaystyle \frac{\sqrt{3}}{2}13\displaystyle \frac{1}{\sqrt{3}}
45∘\displaystyle 45^\circπ4\displaystyle \frac{\pi}{4}22\displaystyle \frac{\sqrt{2}}{2}22\displaystyle \frac{\sqrt{2}}{2}1\displaystyle 1
60∘\displaystyle 60^\circπ3\displaystyle \frac{\pi}{3}32\displaystyle \frac{\sqrt{3}}{2}12\displaystyle \frac{1}{2}3\displaystyle \sqrt{3}
90∘\displaystyle 90^\circπ2\displaystyle \frac{\pi}{2}1\displaystyle 10\displaystyle 0Undefined
Curriculum & Revision Note

About These Trigonometry Formulas

Trigonometry is one of the most formula-dense topics in secondary school and pre-university maths. This sheet brings together SOHCAHTOA ratios for right-angled triangles, the Sine Rule and Cosine Rule for any triangle, non-right triangle area (1/2)·ab·sin(C), exact non-calculator values for 30°, 45°, 60°, and key identities like sin²(θ) + cos²(θ) = 1.

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