The “Co” in Cofunction identities is short for complementary. It shows the relationship between sine & cosine, tangent & cotangent, and secant & cosecant. A particular angle’s trig function is equal to the the angle’s complement’s cofunction. Note that \(\frac{\pi}{2}\) radians \(= 90^{\circ}\)
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 \( \displaystyle \frac{\tan^{2}}{\csc\left(\frac{\pi}{2}-x\right)+1} \)
See Related Pages\(\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)
\(\bullet\text{ Andymath Homepage}\)
