Derivation of the Cofunction Trigonometric Identities

Pronunciation: /ˌdɛɹɪˈveɪʃən ʌv ðə koʊˈˈfʌŋkʃən ˌtrɪɡ ə nəˈmɛ trɪk ɪˈdɛn tɪ 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.

Cite this article as:

McAdams, David E. Derivation of the Cofunction Trigonometric Identities. 7/4/2018. 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

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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