13-3 Trig Functions of Any Angle
For any angle θ (theta) with point (x,y) on its terminal side (or terminal side passing through (x,y)):
sin θ = y
r
cos θ = x
r
tan θ = y
x
x≠0
csc θ = r
y
sec θ = r
x
cot θ = x
y
y≠0
x ≠0
y≠0
r  x2  y2
Signs:
Since “r” is always _________________
the sign of a trig function
depends on x and y.
Sign of :
cos
depends on → ___
sin
depends on → ___
tan
depends on → _____
______
Ex
1: Find the values of the 6 trig functions of θ in standard position passing through the point (8,-15)
sin θ =
r=
y
cos θ =
tan θ =
x
cot θ =
sec θ =
csc θ =
(8,-15)
Quadrantal Angles
Ex 2: Find the 6 trig functions of 180˚
sin 180˚ =
cos 180˚=
tan 180˚=
cot 180˚=
sec 180˚=
csc 180˚=
Finding Reference Angles (θ′ - called theta “prime”)
-
__________________
-
_____________
-
the angle made with the _____________
0˚< θ < 90˚
QI
θ′ =
90˚< θ < 180˚
QII
180˚< θ < 270˚
QIII
θ′ =
θ′ =
-sketch the angle in the correct quadrant
-find the angle made with the x axis
Ex 3: Find the measure of the reference angle for the following:
60˚
140˚
-140˚
320˚
-510˚
225.7˚
270˚< θ < 360˚
QIV
θ′ =
Using Reference Angles to EvaluateTrigonometric Functions
Evaluating the exact value of sin 315˚
- sketch angle (use a ________________ angle if necessary)
- find the _________________ angle and sketch the ___________________ triangle (special ∆)
- find the value of the function and check the sign of your answer
cos 120˚
tan (-120˚)
sec 390˚
csc (-225˚)