Derivation of the Cofunction Trigonometric Identities

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

StepImageEquationDiscussion
1 sin theta = A/C. and cos theta = B/C.. This is the definition of sine and cosine using the angle θ.
2 sin phi = B/C. and cos phi = A/C.. This is the definition of sine and cosine using the angle θ.
3 sin theta = cos phi. and cos theta = sin phi.. This is the definition of sine and cosine using the angle φ.
4 sin theta = cos phi. and cos theta = sin phi. Apply the transitive property of equality to equate sin θ with cos φ and cos θ with sin φ.
5 α + β + γ = &pi This is the Angle Sum Theorem.
6 theta + phi + pi/2 = pi Use the subsitution property of equality to substitute θ for α, φ for β and pi/2 for γ.
7 theta + phi + pi/2 - pi/2 = pi - pi/2 Apply additive property of equality to add -pi/2 to both sides of the equation.
8 theta + phi = pi/2 Simplify the equation by combining the constants on both sides of the equation.
9 theta + phi - theta = pi/2 - theta Apply the addition property of equality to add to both sides of the equation.
10 phi = pi/2 - theta Cancel θ - θ on the left side of the equation.
11 sin theta = cos(pi/2 - theta) and sin(pi/2 - theta) / cos(pi/2 - theta) = cos theta / sin theta implies tan(pi/2 - theta) = cot(theta) Take the equations from step 4 and apply the subsitution property of equality witht the equations from step 10. These are the first two cofunction identities.
12 sin(pi/2 - theta) / cos(pi/2 - theta) = cos theta / sin theta implies tan(pi/2 - theta) = cot(theta) Use the equations from step 11, and the defintions of tangent and cotangent to get the tangent identity.
13 cos(pi/2 - theta) / sin(pi/2 - theta) = sin theta / cos theta implies cot(pi/2 - theta) = tan theta Use the equations from step 11, and the defintions of tangent and cotangent to get the cotangent identity.
14 1 / sin(pi/2 - theta) = 1 / cos( theta) implies csc(pi/2 - theta) = sec(theta) Use the equations from step 11, and the defintions of tangent and cotangent to get the cosecant identity.
15 1 / cos(pi/2 - theta) = 1 / sin( theta) implies sec(pi/2 - theta) = csc(theta) 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. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/t/ti_cofunction.html.

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Revision History

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.)

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