Unit Circle

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Notes

Unit Circle


Unit Circle with Blank Spaces For Students to Practice

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Printable Blank Unit Circle
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Practice Problems

Find the following trig values on the unit circle.

\(\textbf{1)}\) \(\sin{\displaystyle\frac{2\pi}{3}}\)


\(\textbf{2)}\) \(\sin{45^{\circ}}\)


\(\textbf{3)}\) \(\sin{30^{\circ}}\)


\(\textbf{4)}\) \(\cos{\displaystyle\frac{\pi}{6}}\)


\(\textbf{5)}\) \(\tan{210^{\circ}}\)


\(\textbf{6)}\) \(\tan{\displaystyle\frac{4\pi}{3}}\)


\(\textbf{7)}\) \(\sin{-60^{\circ}}\)


\(\textbf{8)}\) \(\cos{-45^{\circ}}\)


\(\textbf{9)}\) \(\tan{90^{\circ}}\)


\(\textbf{10)}\) \(\sin{\displaystyle\frac{5\pi}{4}}\)


\(\textbf{11)}\) \(\sin{150^{\circ}}\)


\(\textbf{12)}\) \(\tan{225^{\circ}}\)


\(\textbf{13)}\) \(\tan{150^{\circ}}\)


\(\textbf{14)}\) \(\sin{\displaystyle\frac{3\pi}{4}}\)


\(\textbf{15)}\) \(\sin{-210^{\circ}}\)


\(\textbf{16)}\) \(\cos{210^{\circ}}\)


\(\textbf{17)}\) \(\tan{180^{\circ}}\)


\(\textbf{18)}\) \(\sin{90^{\circ}}\)


\(\textbf{19)}\) \(\cos{-\displaystyle\frac{2\pi}{3}}\)


\(\textbf{20)}\) \(\sin{300^{\circ}}\)


\(\textbf{21)}\) \(\tan{-45^{\circ}}\)


\(\textbf{22)}\) \(\tan{-\displaystyle\frac{\pi}{3}}\)


\(\textbf{23)}\) \(\cos{\displaystyle\frac{3\pi}{4}}\)


\(\textbf{24)}\) \(\tan{\displaystyle\frac{5\pi}{3}}\)


\(\textbf{25)}\) \(\cos{-\displaystyle\frac{\pi}{6}}\)


\(\textbf{26)}\) \(\cos{0^{\circ}}\)


\(\textbf{27)}\) \(\cos{315^{\circ}}\)


\(\textbf{28)}\) \(\tan{\displaystyle\frac{\pi}{3}}\)


\(\textbf{29)}\) \(\tan{0\pi}\)


\(\textbf{30)}\) \(\tan{-\displaystyle\frac{7\pi}{6}}\)


\(\textbf{31)}\) \(\tan{\displaystyle\frac{2\pi}{3}}\)


\(\textbf{32)}\) \(\sin{210^{\circ}}\)


\(\textbf{33)}\) \(\cos{90^{\circ}}\)


\(\textbf{34)}\) \(\tan{135^{\circ}}\)


\(\textbf{35)}\) \(\tan{315^{\circ}}\)


\(\textbf{36)}\) \(\cos{-\displaystyle\frac{7\pi}{6}}\)


\(\textbf{37)}\) \(\cos{150^{\circ}}\)


\(\textbf{38)}\) \(\cos{-\displaystyle\frac{\pi}{3}}\)


\(\textbf{39)}\) \(\sin{-45^{\circ}}\)


\(\textbf{40)}\) \(\cos{180^{\circ}}\)


\(\textbf{41)}\) \(\cos{\displaystyle\frac{5\pi}{4}}\)


\(\textbf{42)}\) \(\tan{-30^{\circ}}\)


\(\textbf{43)}\) \(\cos{240^{\circ}}\)


\(\textbf{44)}\) \(\sin{-120^{\circ}}\)


\(\textbf{45)}\) \(\sin{270^{\circ}}\)


\(\textbf{46)}\) \(\cos{300^{\circ}}\)


\(\textbf{47)}\) \(\sin{315^{\circ}}\)


\(\textbf{48)}\) \(\sin{240^{\circ}}\)


\(\textbf{49)}\) \(\tan{30^{\circ}}\)


\(\textbf{50)}\) \(\tan{45^{\circ}}\)


\(\textbf{51)}\) \(\sin{\pi}\)


\(\textbf{52)}\) \(\sin{-30^{\circ}}\)


\(\textbf{53)}\) \(\tan{-120^{\circ}}\)


\(\textbf{54)}\) \(\tan{270^{\circ}}\)


\(\textbf{55)}\) \(\cos{45^{\circ}}\)


\(\textbf{56)}\) \(\sin{0^{\circ}}\)


\(\textbf{57)}\) \(\cos{60^{\circ}}\)


\(\textbf{58)}\) \(\cos{270^{\circ}}\)


\(\textbf{59)}\) \(\sin{\displaystyle\frac{\pi}{3}}\)


\(\textbf{60)}\) \(\cos{\displaystyle\frac{2\pi}{3}}\)


\(\textbf{61)}\) \(\sin{120^{\circ}}\)


\(\textbf{62)}\) \(\cos{30^{\circ}}\)


\(\textbf{63)}\) \(\tan{240^{\circ}}\)


\(\textbf{64)}\) \(\sin{225^{\circ}}\)


\(\textbf{65)}\) \(\sin{135^{\circ}}\)


\(\textbf{66)}\) \(\cos{-120^{\circ}}\)


\(\textbf{67)}\) \(\tan{-60^{\circ}}\)


\(\textbf{68)}\) \(\cos{135^{\circ}}\)


\(\textbf{69)}\) \(\tan{300^{\circ}}\)


\(\textbf{70)}\) \(\cos{-30^{\circ}}\)


\(\textbf{71)}\) \(\tan{60^{\circ}}\)


\(\textbf{72)}\) \(\tan{0^{\circ}}\)


\(\textbf{73)}\) \(\tan{-210^{\circ}}\)


\(\textbf{74)}\) \(\tan{120^{\circ}}\)


\(\textbf{75)}\) \(\cos{-210^{\circ}}\)


\(\textbf{76)}\) \(\cos{-60^{\circ}}\)


\(\textbf{77)}\) \(\cos{225^{\circ}}\)


\(\textbf{78)}\) \(\sin{180^{\circ}}\)


\(\textbf{79)}\) \(\sin{60^{\circ}}\)


\(\textbf{80)}\) \(\cos{120^{\circ}}\)


\(\textbf{81)}\) \(\sin{\displaystyle\frac{5\pi}{6}}\)



See Related Pages\(\)

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\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
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\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
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\(\bullet\text{ Coterminal Angles}\)
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\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Reference Angles\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of 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 of a generic triangle\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
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\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) Thumbnail of a generic triangle\(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) Thumbnail of a generic triangle\(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of 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 of a Graph of a Sine Function\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)


In Summary

The unit circle is a fundamental concept in mathematics, specifically in trigonometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The unit circle is often used to help understand and visualize the relationships between angles and their corresponding trigonometric functions.

An important aspect of the unit circle is the concept of reference angles. A reference angle is the acute angle formed between the x-axis and the terminal side of an angle in the coordinate plane. It is used to simplify the calculation of trigonometric functions for angles that are not acute. The points of interest on the unit circle are the intersections with the x- and y- axis and all possible 30-60-90 and 45-45-90 right triangles that have a reference angle in the circle.

The unit circle helps to understand the concept of radians, which is a unit of measurement for angles. One radian is equal to the length of the arc on the unit circle that is formed by the angle, divided by the radius of the circle. This means that the circumference of the unit circle is equal to 2π radians, where π is a mathematical constant approximately equal to 3.14.

The unit circle can also be used to understand the relationships between the trigonometric functions. For example, the sine and cosine functions have a periodic relationship, meaning that they repeat after a certain interval. This interval is known as the period of the function, and for the sine and cosine functions, the period is equal to 2π.

The unit circle can be a lot of fun once you understand it. Try out the many practice problems on this page to get more comfortable with it.

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