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# Sin Pi Over 6

### Sin Pi Over 6 Exact Value

It is essential to learn a tool known as the Unit Circle for trigonometry. This is a circle with the radius #1# with the origin as its center. The coordinates you need to know are the spots on the circle's circumference. When you see a trigonometric function like sine (or sin(#theta#)) or cosine (or cos(#theta#)), it refers to the point on the circle's circumference that intersects the line coming from the origin at a given angle (#theta#) counter-clockwise from the axis between Quadrant I and Quadrant IV of the coordinate plane.

(22.5 ) sin = 1 cos ( 45 ) 2 = 1 2 2 2 = 2 2 2 displaystyle sin(22.5circ )=sqrt rac 1-cos(45circ )2=sqrt rac 1-rac sqrt 2222=rac sqrt 2-sqrt 2222=rac sqrt cos (22.5 ) = 1 plus cos ( 45 ) 2 = 1 + 2 2 2 = 2 + 2 2 displaystyle cos(22.5circ )=sqrt rac 1+cos(45circ ) 2=rac sqrt 1+rac sqrt 22=rac sqrt 2+rac sqrt 22=rac sqrt 2+rac sqrt 22 The cosine half-angle formula is applied repeatedly, resulting in nested square roots that continue in a pattern where each application adds a 2 + displaystyle sqrt 2+cdots to the numerator and the denominator is 2. As an example:

### Sin Pi Over 6 Unit Circle

How do you make use of the unit circle? The right triangle connections sine, cosine, and tangent may be defined using a unit circle. These connections define the relationship between the angles and sides of a right triangle. Assume we have a right triangle with a 30-degree angle and a length of 7 on its longest side, or hypotenuse.

A Word of Advice: Pythagorean Identity According to the Pythagorean Identity, for every real number [latex]t[/latex], [latex]cos 2t+sin 2t=1 [/latex] How To: Find the cosine of [latex]t[/latex] given the sine of some angle [latex]t[/latex] and its quadrant position. In the Pythagorean Identity, substitute the known value of [latex]sin left(t ight)[/latex]. Calculate [latex]cos left(t ight)[/latex]. Select the answer with the proper sign for the x-values in the quadrant containing [latex]t[/latex].

#pi/6# denotes the angle in radians, an auxiliary unit of measurement for angles (#pi# rad = 180). The location on the unit circle where this line intersects is (#sqrt(3)/2#, #1/2#). Finally, the function sin(#theta#) produces a value equal to the point's y-coordinate, yielding an answer of #1/2#. In the future, you should learn all of the key locations on the unit circle, as well as their reference angles, so that you can get these solutions quickly.

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