Cofunction Identities

Cofunction identities show that a trig function of an angle is equal to its cofunction of the complementary angle. The “Co” in cofunction means complementary, connecting sine with cosine, tangent with cotangent, and secant with cosecant. In degrees, complementary angles add to \(90^\circ\), and in radians they add to \(\frac{\pi}{2}\). These identities are useful for solving missing-angle problems and simplifying expressions involving cofunction pairs.

 

Notes

 

Cofunction Identities (Radians)
\(\sin{(x)} = \cos{\left(\frac{\pi}{2} -x\right)}\)
\(\cos{(x)} = \sin{\left(\frac{\pi}{2} -x\right)}\)
\(\sec{(x)} = \csc{\left(\frac{\pi}{2} -x\right)}\)
\(\csc{(x)} = \sec{\left(\frac{\pi}{2} -x\right)}\)
\(\tan{(x)} = \cot{\left(\frac{\pi}{2} -x\right)}\)
\(\cot{(x)} = \tan{\left(\frac{\pi}{2} -x\right)}\)

 

 

Cofunction Identities (Degrees)
\(\sin{(\theta)} = \cos{\left(90 -\theta\right)}\)
\(\cos{(\theta)} = \sin{\left(90 -\theta\right)}\)
\(\sec{(\theta)} = \csc{\left(90 -\theta\right)}\)
\(\csc{(\theta)} = \sec{\left(90 -\theta\right)}\)
\(\tan{(\theta)} = \cot{\left(90 -\theta\right)}\)
\(\cot{(\theta)} = \tan{\left(90 -\theta\right)}\)

 

 

Practice Problems

\(\textbf{1)}\) Solve for x using the cofunction identities
\( \sin{\frac{\pi}{6}}=\cos{x} \)

 

\(\textbf{2)}\) Solve for x using the cofunction identities
\( \csc{\frac{2\pi}{5}}=\sec{x} \)

 

\(\textbf{3)}\) Solve for x using the cofunction identities
\( \tan{\frac{\pi}{5}}=\cot{x} \)

 

\(\textbf{4)}\) Solve for x using the cofunction identities
\( \cos{\frac{8\pi}{9}}=\sin{x} \)

 

\(\textbf{5)}\) Simplify \(\csc{\left(90^\circ-x\right)}\)

 

\(\textbf{6)}\) Solve for x using the cofunction identities
\(\cos{35^\circ}=\sin{x}\)

 

\(\textbf{7)}\) Solve for x using the cofunction identities
\(\tan{22^\circ}=\cot{x}\)

 

\(\textbf{8)}\) Solve for x using the cofunction identities
\(\sec{70^\circ}=\csc{x}\)

 

\(\textbf{9)}\) Solve for x using the cofunction identities
\(\sin{42^\circ}=\cos{x}\)

 

\(\textbf{10)}\) Solve for x using the cofunction identities
\(\cot{17^\circ}=\tan{x}\)

 

\(\textbf{11)}\) Solve for x using the cofunction identities
\(\cos{\left(\frac{\pi}{4}\right)}=\sin{x}\)

 

\(\textbf{12)}\) Solve for x using the cofunction identities
\(\sec{\left(\frac{5\pi}{12}\right)}=\csc{x}\)

 

\(\textbf{13)}\) Solve for x using the cofunction identities
\(\cot{\left(\frac{7\pi}{18}\right)}=\tan{x}\)

 

\(\textbf{14)}\) Solve for x using the cofunction identities
\(\csc{\left(\frac{\pi}{9}\right)}=\sec{x}\)

 

\(\textbf{15)}\) Simplify \(\sin{\left(\frac{\pi}{2}-x\right)}\)

 

\(\textbf{16)}\) Simplify \(\tan{\left(\frac{\pi}{2}-x\right)}\)

 

\(\textbf{17)}\) Simplify \(\sec{\left(\frac{\pi}{2}-x\right)}\)

 

\(\textbf{18)}\) Simplify \( \displaystyle \frac{\tan^{2}{x}}{\csc\left(\frac{\pi}{2}-x\right)+1} \)

 

\(\textbf{19)}\) Simplify \(\displaystyle \frac{\cot{x}}{\tan\left(\frac{\pi}{2}-x\right)}\)

 

\(\textbf{20)}\) Simplify \(\displaystyle \frac{\sec{x}}{\csc\left(\frac{\pi}{2}-x\right)}\)

 

See Related Pages\(\)

\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Reference Angles\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for all 6 Trig Functions\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) Thumbnail for Law of Sines\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) Thumbnail for Area os SAS triangles\(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) Thumbnail for Law of Cosines\(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) Thumbnail for Heron's Formula\(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Similar Right Triangles\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) Thumbnail for Graphing Trig Functions\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)
\(\bullet\text{ Andymath Homepage}\)

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