1. 4.4 Trig Functions of any Angle
I. Definitions of Trigonometric Functions. A) B) C)
1) opp = opposite side, adj = adjacent side.

2. 4.4 Trig Functions of any Angle
II. Pythagorean Theorem and Trig Functions.
A) a2 + b2 = c2 for any right triangle.
1) If you know the (x , y) coordinates then you have the
“a” and “b” terms in the Pyth thm.
2) The “a” (or “b”) term is the x (or y) coordinate, and the
other letter is the other coordinate.
3) The “c” term is the hypotenuse [the “r” in the trig]
B) If you know the coordinates for a given angle, you can use
the Pythgorean theorem to find the hypotenuse (the r).
1) Now you can write all the trig functions using x, y, & r.

3. 4.4 Trig Functions of any Angle
III. Evaluating Trigonometric Functions.
A) sine, cosine, and tangent have different signs in
different quadrants.
1) Because the (x , y) points have different signs.
2) Quad II Quad I
sin θ = sin θ = +
cos θ = cos θ = +
tan θ = tan θ = +
Quad III Quad IV
sin θ = sin θ = -
tan θ = tan θ = -

4. 4.4 Trig Functions of any Angle
IV. Finding the Reference Angle & Using a Calculator.
A) Let θ be any angle in standard position. Its reference
angle is the acute angle θ’ formed by the terminal side
of θ and the horizontal axis (the x-axis).
B) Finding the reference angle. [Quadrant I = no change]
1) Quadrant II = π – θ (rad) or ° – θ (deg)
2) Quadrant III = θ – π (rad) or θ – 180° (deg)
3) Quadrant IV = 2π – θ (rad) or ° – θ (deg)

5. 4.4 Trig Functions of any Angle
IV. Finding the Reference Angle & Using a Calculator.
C) Using a calculator to evaluate trig functions.
1) You have to change the MODE to whatever type of
angle measurements you are using (degrees / radians)
2) To find cosecant, secant, and cot, use the x–1 button.
Examples:
a) To evaluate sin (30°) use degree mode (answer = .5)
b) To evaluate cos (2π/3) use radian mode (answer = –.5)
c) To evaluate tan (-4) use radian mode (answer = –1.158)
d) To evaluate cot (410°) type tan(410°)–1 (answer = .839)
HW: page 318 #1–87 every other odd problem (1, 5, 9, 13 …)