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Find the value of sec(ฮธ) at ฮธ = ฯ€/3 on the unit circle

Find the value of sec(ฮธ) at ฮธ = ฯ€/3 on the unit circle

To find the value of secโก(ฮธ) at ฮธ=ฯ€3 on the unit circle, we first find the cosine of the angle:

cosโก(ฯ€3)=12

Then, since secโก(ฮธ) is the reciprocal of cosโก(ฮธ):

secโก(ฯ€3)=1cosโก(ฯ€3)=112=2

Identify the coordinates of points on the unit circle for given angles

Identify the coordinates of points on the unit circle for given angles

For the angle ฮธ=ฯ€6, the point on the unit circle is given by (cosโก(ฯ€6),sinโก(ฯ€6)).

Calculate these values:

cosโก(ฯ€6)=32

sinโก(ฯ€6)=12

Therefore, the coordinates are:

(32,12)

Find the equation of a tangent to the unit circle at a given point

Find the equation of a tangent to the unit circle at a given point

To find the equation of a tangent to the unit circle at the point (a,b), we start by noting that the unit circle is defined by:

x2+y2=1

The slope of the radius at (a,b) is \x0cracba, so the slope of the tangent line, being perpendicular to the radius, is:

โˆ’\x0cracab

Using the point-slope form of a line, the equation of the tangent line can be written as:

yโ€“b=โˆ’\x0cracab(xโ€“a)

Simplifying, we get:

bx+ay=1

Find the values of tan(ฮธ) for ฮธ in the unit circle at 0, ฯ€/4, ฯ€/3, and ฯ€/2

Find the values of tan(ฮธ) for ฮธ in the unit circle at 0, ฯ€/4, ฯ€/3, and ฯ€/2

To determine the values of tanโก(ฮธ) for ฮธ in the unit circle at 0, ฯ€4, ฯ€3, and ฯ€2, we evaluate the tangent function at these angles:

For ฮธ=0:

tanโก(0)=0

For ฮธ=ฯ€4:

tanโก(ฯ€4)=1

For ฮธ=ฯ€3:

tanโก(ฯ€3)=3

For ฮธ=ฯ€2:

tanโก(ฯ€2)=undefined

Find the angle ฮธ on the unit circle where the following conditions are met: sinโก(ฮธ)=โˆ’12 and cosโก(ฮธ)=โˆ’32

Find the angle ฮธ on the unit circle where the following conditions are met: sinโก(ฮธ)=โˆ’12 and cosโก(ฮธ)=โˆ’32

To find the angle ฮธ on the unit circle where sinโก(ฮธ)=โˆ’12 and cosโก(ฮธ)=โˆ’32, we need to identify the corresponding angles in degrees.

First, note that sinโก(ฮธ)=โˆ’12 occurs at:

ฮธ=210โˆ˜,330โˆ˜

Next, note that cosโก(ฮธ)=โˆ’32 occurs at:

ฮธ=150โˆ˜,210โˆ˜

The common angle is:

ฮธ=210โˆ˜

Find the value of csc(ฮธ) when ฮธ = ฯ€/6 on the unit circle

Find the value of csc(ฮธ) when ฮธ = ฯ€/6 on the unit circle

To find the value of csc(ฮธ) when ฮธ=ฯ€6 on the unit circle, we first find the sine of ฮธ:

sinโก(ฯ€6)=12

Since csc(ฮธ)=1sinโก(ฮธ), we have:

csc(ฮธ)=1sinโก(ฯ€6)=112=2

Find the exact value of cos(pi/9) using the unit circle and trigonometric identities

Find the exact value of cos(pi/9) using the unit circle and trigonometric identities

To find the exact value of cosโก(ฯ€9), we can use the triple-angle identity for cosine:

\n

cosโก(3ฮธ)=4cos3โก(ฮธ)โ€“3cosโก(ฮธ)

\n

Letting ฮธ=ฯ€9, we get:

\n

cosโก(ฯ€3)=4cos3โก(ฯ€9)โ€“3cosโก(ฯ€9)

\n

Since cosโก(ฯ€3)=12, substituting in we have:

\n

12=4cos3โก(ฯ€9)โ€“3cosโก(ฯ€9)

\n

Let x=cosโก(ฯ€9), then the equation becomes:

\n

12=4x3โ€“3x

\n

Multiplying through by 2 to clear the fraction:

\n

1=8x3โ€“6x

\n

This is a cubic equation that can be solved for x=cosโก(ฯ€9) using numerical methods or by recognizing that:

\n

cosโก(ฯ€9)=6+24

Evaluate the integral of sec(x) along the unit circle

Evaluate the integral of sec(x) along the unit circle

To evaluate the integral of secโก(x) along the unit circle, we consider the parametrization of the unit circle. The unit circle can be parametrized as x=cosโก(ฮธ) and y=sinโก(ฮธ), where ฮธ ranges from 0 to 2ฯ€.

The integral to evaluate becomes:

โˆซ02ฯ€secโก(cosโก(ฮธ))dฮธdฮธ dฮธ

We need to express secโก(cosโก(ฮธ)) in terms of ฮธ. However, since secโก(x) is not straightforward to integrate on the unit circle, it is more practical to use a different approach, often involving complex analysis or residue theorem.

Prove that sin(ฯ€/6) using the unit circle

Prove that sin(ฯ€/6) using the unit circle

To prove that sinโก(ฯ€6) using the unit circle, we start by locating the angle ฯ€6 on the unit circle.

The angle ฯ€6 corresponds to 30 degrees.

Using the unit circle, we see that the coordinates for this angle are (32,12).

Since the sine of an angle is the y-coordinate on the unit circle, we have:

sinโก(ฯ€6)=12

Determine the values of cos(ฮธ) and sin(ฮธ) using the unit circle when 0 โ‰ค ฮธ โ‰ค 2ฯ€ and ฮธ is a solution to the equation tan(ฮธ) = โˆš3

Determine the values of cos(ฮธ) and sin(ฮธ) using the unit circle when 0 โ‰ค ฮธ โ‰ค 2ฯ€ and ฮธ is a solution to the equation tan(ฮธ) = โˆš3

The equation tanโก(ฮธ)=3 implies that:

tanโก(ฮธ)=sinโก(ฮธ)cosโก(ฮธ)=3

This happens at ฮธ=ฯ€3 and ฮธ=4ฯ€3 within the interval 0โ‰คฮธโ‰ค2ฯ€.

At ฮธ=ฯ€3:

cosโก(ฯ€3)=12,sinโก(ฯ€3)=32

At ฮธ=4ฯ€3:

cosโก(4ฯ€3)=โˆ’12,sinโก(4ฯ€3)=โˆ’32

Thus, the values are:

ฮธ=ฯ€3:cosโก(ฮธ)=12,sinโก(ฮธ)=32

ฮธ=4ฯ€3:cosโก(ฮธ)=โˆ’12,sinโก(ฮธ)=โˆ’32

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