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\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}}\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}}Quadrantal Angles
0^\circ, 90^\circ, 180^\circ,270^\circ,360^\circ
Reciprocal Identities
\sin\theta=\frac{1}{\csc\theta}\hspace{2mm}\csc\theta=\frac{1}{\sec\theta}\hspace{2mm}\tan\theta=\frac{1}{\cot\theta}\csc\theta=\frac{1}{\sin\theta}\hspace{2mm}\sec\theta=\frac{1}{\cos\theta} \hspace{2mm} \cot\theta=\frac{1}{\tan\theta}Pythagorean Identities
1. \sin^2\theta+\cos^2\theta=1 \\2.\tan^2\theta+1=\sec^2\theta \\3.\cot^2\theta+1=\csc^2\theta
The Pythagorean Identities can be rewritten in the following way:
1.\cos^2\theta=1-\sin^2\theta\\2.\sec^2\theta-1=tan^2\theta\\3.\csc^2\theta-\cot^2\theta=1
For more information on the Pythagorean identities and how they can be used, click here.
Quotient Identities
\tan\theta=\frac{sin\theta}{\cos\theta}\hspace{5mm}\cot\theta=\frac{\cos\theta}{\sin\theta}*Trick to remember the signs in each quadrant
Range of Trig Functions
\sin\theta \hspace{2mm} \textnormal{and} \hspace{2mm} \cos\theta \hspace{2mm} [-1,1]\tan\theta \hspace {2mm} \textnormal{and} \cot\theta \hspace{2mm} (-\infty,\infty)\csc\theta \hspace{2mm} \textnormal{and} \hspace{2mm} \sec\theta \hspace{2mm} (-\infty,\infty)\cup[1,\infty)Cofunction Identities
\sin\theta=\cos(90^\circ-\theta) \\\tan\theta=\cot(90^\circ-\theta) \\\sec\theta=\csc(90^\circ-\theta)
Formula for Arc Length
\textnormal{S}=\textnormal{r}\theta*Theta is in radians!
r = radius, S = the arc length
Formula for Area of a Sector
\textnormal{A}=\frac{1}{2}\textnormal{r}^2\thetar = radius, A = area
Linear Speed Formula
\textnormal{V}=\frac{s}{t}\hspace{2mm}\textnormal{or}\hspace{2mm}\textnormal{V}=\textnormal{r}\omegaAngular Speed Formula
\omega=\frac{\theta}{t}*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(-\theta)=\cos\theta \\ \sec(-\theta)=\sec\theta
Even functions are symmetric with the y-axis.
Odd Functions
\sin(-\theta)=-\sin\theta\\\csc(-\theta)=-\csc\theta\\\tan(-\theta)=-\tan\theta\\\cot(-\theta)=-\cot\theta
Odd functions are symmetric with the origin.
Cofunction Identities
\cos(\frac{\pi}{2}-\theta)=\sin\theta,\hspace{1mm}\sin(\frac{\pi}{2}-\theta)=\cos\theta\sec(\frac{\pi}{2}-\theta)=\csc\theta,\hspace{1mm}\csc(\frac{\pi}{2}-\theta)=\sec\theta\tan(\frac{\pi}{2}-\theta)=\cot\theta,\hspace{1mm}\cot(\frac{\pi}{2}-\theta)=\tan\thetaSum and Difference Identities for Cosine
\cos(\textnormal{A+B) = cosAcosB - sinAsinB}
\\\cos(\textnormal{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
\textnormal{sin(A+B) = sinAcosB + cosAsinB}\\\textnormal{sin(A-B) = sinAcosB - cosAsinB}Sum and Difference Identities for Tangent
\tan(\textnormal{A+B)}=\frac{\tan\textnormal{A + tanB}}{1-(\textnormal{tanA}\cdot\textnormal{tanB)}}\tan(\textnormal{A-B)}=\frac{\tan\textnormal{A - tanB}}{1+(\textnormal{tanA}\cdot\textnormal{tanB)}}Double Angle Formulas
Cosine
1. \cos(2\textnormal{A})=\cos^2\textnormal{A}-\sin^2\textnormal{A}2. \cos(2\textnormal{A})=2\cos^2\textnormal{A}-\sin^2\textnormal{A}3. \cos(2\textnormal{A})=1-\sin^2\textnormal{A}Sine
4.\sin(2\textnormal{A})=2\sin\textnormal{A}\cos\textnormal{A}Tangent
5.\tan(2\textnormal{A})=\frac{2\tan\textnormal{A}}{1-\tan^2\textnormal{A}}6.\tan(2\textnormal{A})=\frac{\sin(2\textnormal{A})}{\cos(2\textnormal{A})}Product-to-Sum Identities
1.\textnormal{cosAcosB = }\frac{1}{2}[\textnormal{cos(A + B) + cos(A - B)}]2. \textnormal{sinAsinB = }\frac{1}{2}[\textnormal{cos(A - B) - cos(A + B)}]3. \textnormal{sinAcosB = }\frac{1}{2}[\textnormal{sin(A + B) + sin(A - B)]}4. \textnormal{cosAsinB = }\frac{1}{2}[\textnormal{sin(A + B) - sin(A-B)]}Sum-to-Product Identities
1. \textnormal{sinA+sinB = 2sin}(\frac{\textnormal{A}+\textnormal{B}}{2})\cos(\frac{\textnormal{A}-\textnormal{B}}{2})2. \textnormal{sinA-sinB = 2cos}(\frac{\textnormal{A}-\textnormal{B}}{2})\sin(\frac{\textnormal{A}-\textnormal{B}}{2})3. \textnormal{cosA+cosB = 2cos}(\frac{\textnormal{A}+\textnormal{B}}{2})\cos(\frac{\textnormal{A}-\textnormal{B}}{2})4. \textnormal{cosA-cosB = -2sin}(\frac{\textnormal{A}+\textnormal{B}}{2})\sin(\frac{\textnormal{A}-\textnormal{B}}{2})For a helpful video on sum to product identities, click here.
Half-Angle Identities
\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos\textnormal{x}}{2}}\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos\textnormal{x}}{2}}\tan\frac{x}{2}=\pm\sqrt{\frac{1-\cos\textnormal{x}}{1+\cos\textnormal{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!