Derivation of the Cofunction Trigonometric Identities

Pronunciation: /ˌdɛɹ.ɪˈveɪ.ʃən ʌv ðə koʊˈfʌŋk.ʃən ˌtrɪɡ.ə.nəˈmɛ.trɪk aɪˈdɛn.tə.tiz/ Explain

Step	Image	Equation	Discussion
1		and .	This is the definition of sine and cosine using the angle θ.
2		and .	This is the definition of sine and cosine using the angle θ.
3		and .	This is the definition of sine and cosine using the angle φ.
4		and	Apply the transitive property of equality to equate sin θ with cos φ and cos θ with sin φ.
5		α + β + γ = π	This is the Angle Sum Theorem.
6			Use the subsitution property of equality to substitute θ for α, φ for β and for γ.
7			Apply additive property of equality to add - to both sides of the equation.
8			Simplify the equation by combining the constants on both sides of the equation.
9			Apply the addition property of equality to add -θ to both sides of the equation.
10			Cancel θ - θ on the left side of the equation.
11		and	Take the equations from step 4 and apply the subsitution property of equality with the equations from step 10. These are the first two cofunction identities.
12			Use the equations from step 11, and the defintions of tangent and cotangent to get the tangent identity.
13			Use the equations from step 11, and the defintions of tangent and cotangent to get the cotangent identity.
14			Use the equations from step 11, and the defintions of tangent and cotangent to get the cosecant identity.
15			Use the equations from step 11, and the defintions of tangent and cotangent to get the secant identity.

References

1. McAdams, David E.. All Math Words Dictionary, derivation. 2nd Classroom edition 20150108-4799968. pg 58. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Derivation of the Cofunction Trigonometric Identities. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/d/dti_cofunction.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/20/2019: Updated equations and expressions to the new format (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
4/29/2011: Initial version. (McAdams, David E.)