Download Trig Test #1: Concepts to Review Page 1 • angular measure, degree

Trig Test #1: Concepts to Review
Page 1
• angular measure, degree, radians, arc length formula
Basic relation: 180 degrees = π radians .
Arc Length: s = rθ.
• right triangle trigonometry, Pythagorean theorem (acute angles)
opp
hyp
θ
adj
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
hypotenuse
opposite
hypotenuse
sec θ =
adjacent
adjacent
cot θ =
opposite
csc θ =
• special triangles (4.2)
The six basic trigonometric functions relate the angle θ to ratios of the length of the sides of the right triangle.
For certain triangles, the trig functions of the angles can be found geometrically. These special triangles occur
frequently enough that it is expected that you learn the value of the trig functions for the special angles.
A 45-45-90 Triangle
Consider the square given below.
1
π/4
1
√
2
1
π/4
1
The angle here must be π/4 radians, since this triangle is half of a square of side length 1.
Now, we can write down all the trig functions for an angle of π/4 radians = 45 degrees:
π 1
opposite
=√
4
hypotenuse
2
π adjacent
1
=
=√
cos
4
hypotenuse
2
π opposite
1
tan
=
= =1
4
adjacent
1
sin
=
csc
π =
1
=
√
2
4
sin
π √
1
sec
=
=
2
4
cos π4
π 1
=1
cot
=
4
tan π4
π
4
Trig Test #1: Concepts to Review
Page 2
A 30-60-90 Triangle
Consider the equilateral triangle given below. Geometry allows us to construct a 30-60-90 triangle:
We can now determine the six trigonometric functions at two more angles!
π
60◦ = radians:
3
√
2
3
π/3
1
√
opposite
3
sin
=
=
3
hypotenuse
2
π adjacent
1
cos
=
=
3
hypotenuse
2
√
π opposite
3 √
tan
=
=
= 3
3
adjacent
1
π 30o =
csc
sec
cot
π 3
π 3
π 3
=
1
sin
π
3
2
=√
3
=
1
=2
cos π3
=
1
1
√
π =
tan 3
3
π
radians:
6
2
1
π/6
√
3
opposite
1
=
6
hypotenuse
2
√
π adjacent
3
cos
=
=
6
hypotenuse
2
π opposite
1
tan
=
=√
6
adjacent
3
sin
π =
csc
π 6
1
sin
π
3
=2
1
2
=√
π
6
cos 3
3
√
π 1
3 √
=
cot
=
= 3
π
6
1
tan 3
sec
π =
=
Trig Test #1: Concepts to Review
Page 3
• initial side, terminal side, vertex, standard position, coterminal angles, quadrantal angles, quadrants, CAST
(4.3)
II
6
S A
T C
III
I
P (x, y)
@
I
@
-
y
6
@ θ
x
-
IV
If we label the point at the end of the terminal side as P (x, y), and if we let r =
the following relationships between the six trig functions and our diagram:
p
x2 + y 2 , we can construct
x
y
y
,
sin θ = ,
tan θ = , x 6= 0
r
r
x
r
r
x
csc θ = , y =
6 0,
sec θ = , x 6= 0,
cot θ = , y 6= 0
y
x
y
cos θ =
• the unit circle: the above relations with r = 1 produces the unit circle. The coordinates around the unit circle
satisfy (x, y) = (cos θ, sin θ).
• a periodic function f satisfies f (x) = f (x + c) where c is the smallest such number and c is called the period.