Download Section 2.1 - Trig. Functions of Acute Triangles.notebook

Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
Section 2.1:
Trigonometric Functions of Acute Angles
Right-Triangle-Based Definition of the Trig Functions:
In Chapter 1, we used angles in standard position to define the trig
functions. There is another way to approach them: as ratios of the
lengths of the sides of right triangles.
y = side opposite angle A
x = side adjacent angle A
r = hypotenuse
Right-Triangle-Based Definitions of Trig Functions
For any acute angle A in standard position,
sin A = y = opposite
r hypotenuse
csc A = r = hypotenuse
y
opposite
cos A = x = adjacent
r hypotenuse
sec A = r = hypotenuse
x
adjacent
tan A = y = opposite
x adjacent
cot A = x = adjacent
y
opposite
Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
EXAMPLE 1
Finding trigonometric function values of an acute angle
Find the sine, cosine, and tangent values for
angles A and B in the right triangle
8
C
B
15
17
A
Cofunctions
In example 1, you may have noticed that sin A = cos B
and cos A = sin B. Such relationships are always true
for the two acute angles of a right triangle.
(Note: We ALWAYS use C to be the right angle in an ABC triangle.) B
a
C
sin A =
a
c
= cos B
tan A =
a
b
= cot B
sec A =
c
b
= csc B
c
b
A
Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
CofunctionIdentities:
Foranyacuteangle A,
sinA=cos(90°‐A);cosA=sin(90°‐A)
cscA=sec(90°‐A);secA=csc(90°‐A)
tanA=cot(90°‐A);cotA=tan(90°‐A)
EXAMPLE 2
Write each function in terms of its cofunctions.
(a) cos 41°
(b) tan 13°
(c) sec 84°
EXAMPLE 3
Solving Equations Using the Cofunction Identities
Find one solution for each equation. Assume
all angles involved are acute angles.
cos(3θ + 5°) = sin(6θ + 10°)
tan(4θ - 1°) = cot(5θ + 22°)
Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
Comparing Function Values
of Acute Triangle
θ
θ
θ
As angle θ increases from 0o to 90o, sin θ ___________. As angle θ increases from 0o to 90o, cos θ ___________.
As angle θ increases from 0o to 90o, tan θ ___________.
EXAMPLE 4
Tell whether each statement is true or false.
(a) sin 43° > sin 22°
(b) cos 71° ≤ cos 86°
Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
Special Right Triangles
30o - 60o - 90o
θ
30o
45o
60o
45o - 45o - 90o
sin θ cos θ tan θ csc θ sec θ cot θ
Section 2.1 ­ Trig. Functions of Acute Triangles.notebook
February 08, 2013
Drill Quiz #2
Quadrantal and Special Right Angles!