Find the value of sec(ฮธ) at ฮธ = ฯ/3 on the unit circle
To find the value of
Then, since
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Find the value of sec(ฮธ) at ฮธ = ฯ/3 on the unit circle
To find the value of
Then, since
Identify the coordinates of points on the unit circle for given angles
For the angle
Calculate these values:
Therefore, the coordinates are:
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
The slope of the radius at
Using the point-slope form of a line, the equation of the tangent line can be written as:
Simplifying, we get:
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
For
For
For
For
Find the angle
To find the angle
First, note that
Next, note that
The common angle is:
Find the value of csc(ฮธ) when ฮธ = ฯ/6 on the unit circle
To find the value of
Since
Find the exact value of cos(pi/9) using the unit circle and trigonometric identities
To find the exact value of
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Letting
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Since
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Let
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Multiplying through by 2 to clear the fraction:
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This is a cubic equation that can be solved for
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Evaluate the integral of sec(x) along the unit circle
To evaluate the integral of
The integral to evaluate becomes:
We need to express
Prove that sin(ฯ/6) using the unit circle
To prove that
The angle
Using the unit circle, we see that the coordinates for this angle are
Since the sine of an angle is the y-coordinate on the unit circle, we have:
Determine the values of cos(ฮธ) and sin(ฮธ) using the unit circle when 0 โค ฮธ โค 2ฯ and ฮธ is a solution to the equation tan(ฮธ) = โ3
The equation
This happens at
At
At
Thus, the values are:
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