Trigonometry The Unit Circle. Imagine a circle on the co- ordinate plane, with its center at the...
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Transcript of Trigonometry The Unit Circle. Imagine a circle on the co- ordinate plane, with its center at the...
Trigonometry
The Unit Circle
The Unit Circle
Imagine a circle on the co-ordinate plane, with its center at the origin, and a radius of 1.
Choose a point on the circle somewhere in the first quadrant.
The Unit Circle
Connect the origin to the point, and from that point drop a perpendicular to the x-axis.
This creates a right triangle with hypotenuse of 1.
1
The Unit Circle The length of sides of the triangle are
the x and y co-ordinates of the chosen point.
Applying the definitions of
the trigonometric ratios to
this triangle gives
cos1
xx
x
y1
θ
sin1
yy
The co-ordinates of the chosen point are the cosine and sine of the angle .
This provides a way to define functions sin and cos for all real numbers .
The other trigonometric functions can be defined from these.
The Unit Circle
sin1
yy cos
1
xx
Trigonometric Functions
sin y
cos x
tany
x
1cosec
y
1sec
x
cotx
y
x
y1
θ
cosecant
secant
cotan
Around the Circle
As that point moves around the unit circle into the second, third and fourth quadrants, the new definitions of the trigonometric functions still hold.
Reference Angles
The angles whose terminal sides fall in the 2nd, 3rd, and 4th quadrants will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in 1st quadrant.
The acute angle which produces the same values is called the reference angle.
Second Quadrant
Original angle θ
Reference angle
For an angle , in the second quadrant, the reference angle is
In the second quadrant,
sin is positive
cos is negative
tan is negative
Third Quadrant
Original angle θ
Reference angle
For an angle , in the third quadrant, the reference angle is –
In the third quadrant,
sin is negative
cos is negative
tan is positive
Fourth Quadrant
Original angle θ
Reference angle
For an angle , in the fourth quadrant, the reference angle is 2
In the fourth quadrant,
sin is negative
cos is positive
tan is negative
All Students Take Care
AllStudents
Take Care
Use the phrase “All Students Take Care” to remember the signs of the trigometric functions in the different quadrants.
CS A
T Cine is pS ositive ll posA itive
os posC itivean is pT ositive
Examples
Find sin240° in surd form.CS A
T C– Draw the angle on the unit circle
– In the 3rd quadrant sine is negative
– Find the angle to nearest x-axis 60º
3sin 240
2 Page 9 of tables
3sin 60
2
Examples
cosθ = – 0·5. Find the two possible values of θ, where 0º ≤ θ ≤ 360°.
S A
T C
60ºcosA = 0·5
cos is negative in two quadrants
2nd
3rd 180º + 60º 240º
180º – 60º 120º