Trig equations. How do I found where the point 1/3 is on unit circle

Question

The equation is tan(thetha)= 1/3 . [0,360)degrees

 

How am I supposed to know how to find the angle? Is there some equation to do? I know the unit circle by heart so its what im using, but it doesnt show this point on it. 

Answer 1

On the unit circle, x = cox(θ) and y = sin(θ)

 

The unit circle is x² + y² =1

 

tan(θ) = sin(θ)/cos(θ) = y/x = 1/3

y = x/3

 

Put these values into the equation for the unit circle.

 

x² + y² = x² + (x/3)² = (1 + 1/9)x²  10/9x² = 1

x² = 9/10 => x = ±3/√(10)

 

y² = 1 - x² = 1 - 9/10 = 1/10

y = ±1/√(10)

 

So the points on the unit circle with tan(θ) = 1/3 are

(x, y) = (cos(θ), sin(θ)) = (3/√(10), 1/√(10)) and

(x, y) = (cos(θ), sin(θ)) = (-3/√(10), -1/√(10))

 

The angle in the first quadrant, when cos(θ) = 3/√(10),

the angle can represented by the arc-cosine function as follows

θ = cos-¹(3/√(10)).  This function should be on a graphing

calculator which will give the value for the angle.

 

The second pair of values given for a solution is in the

third quadrant and represents the angle

180 + θ =180 + cos^-1(3/√(10))

Again a graphing calculator can be used to find its value

 

The angle in the first quadrant can also be found in a table of

sines, cosines, or tangents by finding the value needed and

then seeing what the closest angle is with that value.

Answer 2

Use the inverse tangent function to find your reference angle.  Since your tangent value is positive, you will have 2 angles that will satisfy the equation in the given interval.  The reference angle will be in the 1st quadrant.  The second angle will be in the 3rd quadrant.