Transcript 4.4: Trig of Any Angle - Warren County Public Schools

Objectives:
1. To find trig values of
an angle given any
point on the terminal
side of an angle
2. To find the acute
reference angle of any
angle
Recall that when using the
unit circle to evaluate the
value of a trig function,
cos θ = x and sin θ = y.
What we didn’t point out
is that since the radius
(hypotenuse) is 1, the trig
values are really cos θ =
x/1 and sin θ = y/1.
So what if the radius
(hypotenuse) is not 1?
Let θ be an angle whose terminal side contains
the point (4, 3). Find sin θ, cos θ, and tan θ.
Let θ be an angle whose terminal side contains
the point (4, 3). Find sin θ, cos θ, and tan θ.
Realize that the triangle formed
by the point (4, 3) is similar to a
triangle in the unit circle. To
get to that unit circle triangle,
we would have to scale down
the larger triangle by dividing
by the scale factor. In this case,
that’s 5, the length of the larger
hypotenuse.
Let (x, y) be a point on the terminal side of an
angle θ in standard position with r  x 2  y 2  0
sin  
y
r
csc  
cos  
x
r
r
, x0
x
x
cot   , y  0
y
y
tan   , x  0
x
sec  
r
, y0
y
The previous definitions imply that tan θ and sec
θ are not defined when x = 0. So what values
of θ are we talking about?
They also imply that cot θ and csc θ are not
defined when y = 0. So what values of θ are
we talking about?
Let θ be an angle whose terminal side contains
the point (−2, 5). Find the six trig functions for
θ.
We can find the sign
of a particular trig
function based on
which quadrant
(x, y) lies within.
Given sin θ = 4/5 and tan θ < 0, find cos θ and
csc θ.
When building the
unit circle, for
120° we drew a
triangle with the
x-axis to form a
60° angle. This
60° angle was
the reference
angle for 120°.
If we connect the
trig of any angle
to right triangle
trigonometry,
we need a
reference angle
that is acute to
be able to
evaluate the
function.
Let θ be an angle in standard position. It’s reference
angle is the acute angle θ’ formed by the terminal
side of θ and the x-axis.
 '  180 
 '   180
 '  360  
Let θ be an angle in standard position. It’s reference
angle is the acute angle θ’ formed by the terminal
side of θ and the x-axis.
 '   
 '   
 '  2 
Find the reference angle for each of the
following.
1. 213°
2. 1.7 rad
3. −144°
When your angle is negative or is greater than
one revolution, to find the reference angle,
first find the positive coterminal angle
between 0° and 360° or 0 and 2π.
How Reference Angles Work:
y
sin  
r
sin  ' 
y
r
Same except
maybe a difference
of sign.
To find the value of a
trig function of any
angle:
1. Find the trig value
for the associated
reference angle
2. Pick the correct
sign depending on
where the
terminal side lies
Evaluate:
1. sin 5π/3
2. cos (−60°)
3. tan 11π/6
Let θ be an angle in Quadrant III such that
sin θ = −5/13. Find a) sec θ and b) tan θ using
trig identities.
• Page 294-296
1-11 odd, 18-24 even, 29-35 odd, 46-52 even,
53-69 odd, 78-88 even