WAC TYPE 2: WHAT IS THE RELATIONSHIP BETWEEN THE
SINE AND COSINE FUNCTIONS OF THE ACUTE ANGLES IN A
RIGHT TRIANGLE?
http://padlet.com/wall/8tx4w84jl8sl
IN CASE YOU MISSED IT
THE UNIT CIRCLE
WHAT IS A UNIT CIRCLE?
• The unit circle is a circle with the equation 𝑥 2 + 𝑦 2 = 1
• What do we know about this circle?
THE UNIT CIRCLE
• If this is a circle with radius = 1, label
the coordinates of the points where the
circle crosses the x and y axes.
• Can we label the coordinates of the
other points along the unit circle where
the major angles are?
• Take out your unit circle from the page
protector and fill in the coordinates
according to the activity we just did.
TRIG AND THE UNIT CIRCLE
• Go back in your notes to our table of trig values for the major angles.
Compare these values to the coordinates we just found. What do you notice?!
• The x-coordinate is sin θ and the y-coordinate is cos θ!!!
• An (x,y) ordered pair on the unit circle gives you the sin and cos values, which
will allow you to find other trig function values…THIS IS HUGE!!
REFERENCE ANGLES
• Let Ө be an angle in standard position.
• Its reference angle is the acute angle, Ө’, formed by the terminal side of Ө and
the horizontal axis
• The trig function’s value for Ө is the same as the associated reference angle,
Ө’, the only thing that is different is the sign
TO FIND REFERENCE ANGLES:
Quadrant 2:
Quadrant 3:
 
180  
 
  180
Quadrant 4:
2  
360  
When working in
radians, the reference
angles are the ones with
the same denominator
REDEFINING THE TRIG FUNCTIONS IN THE UNIT
CIRCLE
• Now that we know how the trig functions relate to the coordinates on the
unit circle, we are going to redefine how we find them.
REDEFINING THE TRIG FUNCTIONS IN THE UNIT
CIRCLE
• Now that we know how the trig functions relate to the coordinates on the
unit circle, we are going to redefine how we find them.
sin 𝜃 = 𝑦
csc 𝜃 =
1
1
= ,
sin 𝜃 𝑦
𝑥≠0
cos 𝜃 = 𝑥
sec 𝜃 =
1
1
= ,
cos 𝜃 𝑥
𝑦≠0
sin 𝜃 𝑦
tan 𝜃 =
= ,
cos 𝜃 𝑥
𝑥≠0
cot 𝜃 =
1
cos 𝜃 𝑥
=
= ,
tan 𝜃 sin 𝜃 𝑦
𝑥≠0
ALL STUDENTS TAKE CALCULUS
• Since sine is the y coordinate, it will be positive in the first and second
quadrants.
• Since cosine is the x coordinate, it will be positive in the first and fourth
quadrants.
• What about tangent?
• Or, just remember All Students Take Calculus
• All trig functions are positive in Q1
• Sine (and cosecant) are positive in Q2
• Tangent (and cotangent) are positive in Q3
• Cosine (and secant) are positive in Q4
THE LEFT HAND TRICK
• Imagine your left hand as an axis showing the
first quadrant. Your thumb is the y-axis and your
pinky is the x-axis.
• Imagine the other fingers are our important
angles.
𝑓𝑖𝑛𝑔𝑒𝑟𝑠
2
0°, 360°, 0, 2𝜋
• To find the sine value, hold down the finger that
represents the angle you want. Now count the
number of fingers below the bent one. This goes
under the radical.
• To find the cosine value, do the same as above,
but count the fingers above the bent one.
• To find the tangent value, take the square root
of the bottom fingers over the square root of
the top ones (ignore the 2).