Trigonometry formulas are your best friend in trigonometry. Memorize these to help you solve all trigonometry related problems you may run into.
6 Trig Functions
sin θ = y r or opposite hypotenuse cos θ = x r or adjacent hypotenuse tan θ = y x or opposite adjacent \sin\theta=\frac{y}{r} \hspace{2 mm}\hspace{2mm}\textnormal{or}\hspace{2mm} \frac{\text{opposite}}{\text{hypotenuse}} \hspace {5mm}
\cos\theta=\frac{x}{r}\hspace{2mm}\textnormal{or}\hspace{2mm}\frac{\text{adjacent}}{\text{hypotenuse}}\hspace{5mm}
\tan\theta=\frac{y}{x}\hspace{2mm}\textnormal{or}\hspace{2mm}\frac{\text{opposite}}{\text{adjacent}} sin θ = r y or hypotenuse opposite cos θ = r x or hypotenuse adjacent tan θ = x y or adjacent opposite csc θ = r y or hypotenuse opposite sec θ = r x or hypotenuse a d j a c e n t cot θ = x y or adjacent opposite \csc\theta=\frac{r}{y}\hspace{2mm}\textnormal{or}\hspace{2mm}\frac{\text{hypotenuse}}{\text{opposite}}\hspace{5mm}
\sec\theta=\frac{r}{x}\hspace{2mm}\textnormal{or}\hspace{2mm}\frac{\text{hypotenuse}}{adjacent}\hspace{5mm}
\cot\theta=\frac{x}{y}\hspace{2mm}\textnormal{or}\hspace{2mm}\frac{\text{adjacent}}{\text{opposite}} csc θ = y r or opposite hypotenuse sec θ = x r or a d ja ce n t hypotenuse cot θ = y x or opposite adjacent
Quadrantal Angles
0 ∘ , 9 0 ∘ , 18 0 ∘ , 27 0 ∘ , 36 0 ∘ 0^\circ, 90^\circ, 180^\circ,270^\circ,360^\circ 0 ∘ , 9 0 ∘ , 18 0 ∘ , 27 0 ∘ , 36 0 ∘
Reciprocal Identities
sin θ = 1 csc θ csc θ = 1 sec θ tan θ = 1 cot θ \sin\theta=\frac{1}{\csc\theta}\hspace{2mm}\csc\theta=\frac{1}{\sec\theta}\hspace{2mm}\tan\theta=\frac{1}{\cot\theta} sin θ = csc θ 1 csc θ = sec θ 1 tan θ = cot θ 1 csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ \csc\theta=\frac{1}{\sin\theta}\hspace{2mm}\sec\theta=\frac{1}{\cos\theta} \hspace{2mm} \cot\theta=\frac{1}{\tan\theta} csc θ = sin θ 1 sec θ = cos θ 1 cot θ = tan θ 1
Pythagorean Identities
1. sin 2 θ + cos 2 θ = 1 2. tan 2 θ + 1 = sec 2 θ 3. cot 2 θ + 1 = csc 2 θ 1. \sin^2\theta+\cos^2\theta=1
\\2.\tan^2\theta+1=\sec^2\theta
\\3.\cot^2\theta+1=\csc^2\theta 1. sin 2 θ + cos 2 θ = 1 2. tan 2 θ + 1 = sec 2 θ 3. cot 2 θ + 1 = csc 2 θ The Pythagorean Identities can be rewritten in the following way:
1. cos 2 θ = 1 − sin 2 θ 2. sec 2 θ − 1 = t a n 2 θ 3. csc 2 θ − cot 2 θ = 1 1.\cos^2\theta=1-\sin^2\theta\\2.\sec^2\theta-1=tan^2\theta\\3.\csc^2\theta-\cot^2\theta=1 1. cos 2 θ = 1 − sin 2 θ 2. sec 2 θ − 1 = t a n 2 θ 3. csc 2 θ − cot 2 θ = 1
For more information on the Pythagorean identities and how they can be used, click here .
Quotient Identities
tan θ = s i n θ cos θ cot θ = cos θ sin θ \tan\theta=\frac{sin\theta}{\cos\theta}\hspace{5mm}\cot\theta=\frac{\cos\theta}{\sin\theta} tan θ = cos θ s in θ cot θ = sin θ cos θ
*Trick to remember the signs in each quadrant
Range of Trig Functions
sin θ and cos θ [ − 1 , 1 ] \sin\theta \hspace{2mm} \textnormal{and} \hspace{2mm} \cos\theta \hspace{2mm} [-1,1] sin θ and cos θ [ − 1 , 1 ] tan θ and cot θ ( − ∞ , ∞ ) \tan\theta \hspace {2mm} \textnormal{and} \cot\theta \hspace{2mm} (-\infty,\infty) tan θ and cot θ ( − ∞ , ∞ ) csc θ and sec θ ( − ∞ , ∞ ) ∪ [ 1 , ∞ ) \csc\theta \hspace{2mm} \textnormal{and} \hspace{2mm} \sec\theta \hspace{2mm} (-\infty,\infty)\cup[1,\infty) csc θ and sec θ ( − ∞ , ∞ ) ∪ [ 1 , ∞ )
Cofunction Identities
sin θ = cos ( 9 0 ∘ − θ ) tan θ = cot ( 9 0 ∘ − θ ) sec θ = csc ( 9 0 ∘ − θ ) \sin\theta=\cos(90^\circ-\theta)
\\\tan\theta=\cot(90^\circ-\theta)
\\\sec\theta=\csc(90^\circ-\theta) sin θ = cos ( 9 0 ∘ − θ ) tan θ = cot ( 9 0 ∘ − θ ) sec θ = csc ( 9 0 ∘ − θ )
Formula for Arc Length
S = r θ \textnormal{S}=\textnormal{r}\theta S = r θ
*Theta is in radians!
r = radius, S = the arc length
Formula for Area of a Sector
A = 1 2 r 2 θ \textnormal{A}=\frac{1}{2}\textnormal{r}^2\theta A = 2 1 r 2 θ r = radius, A = area
Linear Speed Formula
V = s t or V = r ω \textnormal{V}=\frac{s}{t}\hspace{2mm}\textnormal{or}\hspace{2mm}\textnormal{V}=\textnormal{r}\omega V = t s or V = r ω
Angular Speed Formula
ω = θ t \omega=\frac{\theta}{t} ω = t θ *Theta is in radians
For more information on the linear and angular speed formulas and examples of how they are used, click here .
Even Functions
cos ( − θ ) = cos θ sec ( − θ ) = sec θ \cos(-\theta)=\cos\theta
\\
\sec(-\theta)=\sec\theta cos ( − θ ) = cos θ sec ( − θ ) = sec θ Even functions are symmetric with the y-axis.
Odd Functions
sin ( − θ ) = − sin θ csc ( − θ ) = − csc θ tan ( − θ ) = − tan θ cot ( − θ ) = − cot θ \sin(-\theta)=-\sin\theta\\\csc(-\theta)=-\csc\theta\\\tan(-\theta)=-\tan\theta\\\cot(-\theta)=-\cot\theta sin ( − θ ) = − sin θ csc ( − θ ) = − csc θ tan ( − θ ) = − tan θ cot ( − θ ) = − cot θ Odd functions are symmetric with the origin.
Cofunction Identities
cos ( π 2 − θ ) = sin θ , sin ( π 2 − θ ) = cos θ \cos(\frac{\pi}{2}-\theta)=\sin\theta,\hspace{1mm}\sin(\frac{\pi}{2}-\theta)=\cos\theta cos ( 2 π − θ ) = sin θ , sin ( 2 π − θ ) = cos θ
sec ( π 2 − θ ) = csc θ , csc ( π 2 − θ ) = sec θ \sec(\frac{\pi}{2}-\theta)=\csc\theta,\hspace{1mm}\csc(\frac{\pi}{2}-\theta)=\sec\theta sec ( 2 π − θ ) = csc θ , csc ( 2 π − θ ) = sec θ tan ( π 2 − θ ) = cot θ , cot ( π 2 − θ ) = tan θ \tan(\frac{\pi}{2}-\theta)=\cot\theta,\hspace{1mm}\cot(\frac{\pi}{2}-\theta)=\tan\theta tan ( 2 π − θ ) = cot θ , cot ( 2 π − θ ) = tan θ
Sum and Difference Identities for Cosine
cos ( A+B) = cosAcosB - sinAsinB cos ( A-B) = cosAcosB + sinAsinB \cos(\textnormal{A+B) = cosAcosB - sinAsinB}
\\\cos(\textnormal{A-B) = cosAcosB + sinAsinB} cos ( A+B) = cosAcosB - sinAsinB cos ( A-B) = cosAcosB + sinAsinB Be sure to note that:
cos(A+B) does not equal cosA + cosB
cos(A-B) does not equal cosA – cosB
For more in depth coverage of the sum and difference identities for cosine and examples of how they are used, click here .
Sum and Difference Identities for Sine
sin(A+B) = sinAcosB + cosAsinB sin(A-B) = sinAcosB - cosAsinB \textnormal{sin(A+B) = sinAcosB + cosAsinB}\\\textnormal{sin(A-B) = sinAcosB - cosAsinB} sin(A+B) = sinAcosB + cosAsinB sin(A-B) = sinAcosB - cosAsinB
Sum and Difference Identities for Tangent
tan ( A+B) = tan A + tanB 1 − ( tanA ⋅ tanB) \tan(\textnormal{A+B)}=\frac{\tan\textnormal{A + tanB}}{1-(\textnormal{tanA}\cdot\textnormal{tanB)}} tan ( A+B) = 1 − ( tanA ⋅ tanB) tan A + tanB tan ( A-B) = tan A - tanB 1 + ( tanA ⋅ tanB) \tan(\textnormal{A-B)}=\frac{\tan\textnormal{A - tanB}}{1+(\textnormal{tanA}\cdot\textnormal{tanB)}} tan ( A-B) = 1 + ( tanA ⋅ tanB) tan A - tanB
Double Angle Formulas
Cosine
1. cos ( 2 A ) = cos 2 A − sin 2 A 1. \cos(2\textnormal{A})=\cos^2\textnormal{A}-\sin^2\textnormal{A} 1. cos ( 2 A ) = cos 2 A − sin 2 A 2. cos ( 2 A ) = 2 cos 2 A − sin 2 A 2. \cos(2\textnormal{A})=2\cos^2\textnormal{A}-\sin^2\textnormal{A} 2. cos ( 2 A ) = 2 cos 2 A − sin 2 A 3. cos ( 2 A ) = 1 − sin 2 A 3. \cos(2\textnormal{A})=1-\sin^2\textnormal{A} 3. cos ( 2 A ) = 1 − sin 2 A Sine
4. sin ( 2 A ) = 2 sin A cos A 4.\sin(2\textnormal{A})=2\sin\textnormal{A}\cos\textnormal{A} 4. sin ( 2 A ) = 2 sin A cos A Tangent
5. tan ( 2 A ) = 2 tan A 1 − tan 2 A 5.\tan(2\textnormal{A})=\frac{2\tan\textnormal{A}}{1-\tan^2\textnormal{A}} 5. tan ( 2 A ) = 1 − tan 2 A 2 tan A 6. tan ( 2 A ) = sin ( 2 A ) cos ( 2 A ) 6.\tan(2\textnormal{A})=\frac{\sin(2\textnormal{A})}{\cos(2\textnormal{A})} 6. tan ( 2 A ) = cos ( 2 A ) sin ( 2 A )
Product-to-Sum Identities
1. cosAcosB = 1 2 [ cos(A + B) + cos(A - B) ] 1.\textnormal{cosAcosB = }\frac{1}{2}[\textnormal{cos(A + B) + cos(A - B)}] 1. cosAcosB = 2 1 [ cos(A + B) + cos(A - B) ] 2. sinAsinB = 1 2 [ cos(A - B) - cos(A + B) ] 2. \textnormal{sinAsinB = }\frac{1}{2}[\textnormal{cos(A - B) - cos(A + B)}] 2. sinAsinB = 2 1 [ cos(A - B) - cos(A + B) ] 3. sinAcosB = 1 2 [ sin(A + B) + sin(A - B)] 3. \textnormal{sinAcosB = }\frac{1}{2}[\textnormal{sin(A + B) + sin(A - B)]} 3. sinAcosB = 2 1 [ sin(A + B) + sin(A - B)] 4. cosAsinB = 1 2 [ sin(A + B) - sin(A-B)] 4. \textnormal{cosAsinB = }\frac{1}{2}[\textnormal{sin(A + B) - sin(A-B)]} 4. cosAsinB = 2 1 [ sin(A + B) - sin(A-B)]
Sum-to-Product Identities
1. sinA+sinB = 2sin ( A + B 2 ) cos ( A − B 2 ) 1. \textnormal{sinA+sinB = 2sin}(\frac{\textnormal{A}+\textnormal{B}}{2})\cos(\frac{\textnormal{A}-\textnormal{B}}{2}) 1. sinA+sinB = 2sin ( 2 A + B ) cos ( 2 A − B ) 2. sinA-sinB = 2cos ( A − B 2 ) sin ( A − B 2 ) 2. \textnormal{sinA-sinB = 2cos}(\frac{\textnormal{A}-\textnormal{B}}{2})\sin(\frac{\textnormal{A}-\textnormal{B}}{2}) 2. sinA-sinB = 2cos ( 2 A − B ) sin ( 2 A − B ) 3. cosA+cosB = 2cos ( A + B 2 ) cos ( A − B 2 ) 3. \textnormal{cosA+cosB = 2cos}(\frac{\textnormal{A}+\textnormal{B}}{2})\cos(\frac{\textnormal{A}-\textnormal{B}}{2}) 3. cosA+cosB = 2cos ( 2 A + B ) cos ( 2 A − B ) 4. cosA-cosB = -2sin ( A + B 2 ) sin ( A − B 2 ) 4. \textnormal{cosA-cosB = -2sin}(\frac{\textnormal{A}+\textnormal{B}}{2})\sin(\frac{\textnormal{A}-\textnormal{B}}{2}) 4. cosA-cosB = -2sin ( 2 A + B ) sin ( 2 A − B )
For a helpful video on sum to product identities, click here .
Half-Angle Identities
sin x 2 = ± 1 − cos x 2 \sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos\textnormal{x}}{2}} sin 2 x = ± 2 1 − cos x cos x 2 = ± 1 + cos x 2 \cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos\textnormal{x}}{2}} cos 2 x = ± 2 1 + cos x tan x 2 = ± 1 − cos x 1 + cos x \tan\frac{x}{2}=\pm\sqrt{\frac{1-\cos\textnormal{x}}{1+\cos\textnormal{x}}} tan 2 x = ± 1 + cos x 1 − cos x Conclusion
Hopefully you find the trigonometry formulas useful going forward! If this article was helpful to you or if you have any questions, comments, or concerns, please leave a comment below!
