Download Precalculus Trigonometry Angles Remember: 360◦ = 2π radians

Precalculus
Trigonometry
Angles
Remember: 360◦ = 2π radians. Therefore,
2π
.
360◦
360◦
• To convert radians to degrees, multiply by
.
2π
• To convert degrees to radians, multiply by
THE UNIT CIRCLE
The unit circle is the circle of radius 1 centered at the origin, (0, 0). Each angle θ corresponds to a point
on the circumference of this circle. The x-coordinate of this point is cos θ, and the y-coordinate of this
point is sin θ.
(sin θ)
y
− 21
√
√
0
4
2 , 2
π
3
π
4
=
θ
3π 4
1
2
π
150 ◦
=
θ=
5π
6
√
=
θ=
7π
6
4
5◦
=
31
4π
3
3π
2
=
240 ◦
θ=
θ = 270◦ =
=
7π 4
θ
=
22
◦
5
−1
√ 2
2
,
−
2
2
5π
3
√ 1
3
,
−
2
2
√
√ 1
3
,
−
2
2
−1
√
√ 0
4
,
−
2
2
−1
− 12
√
=
√
−
√ 3
1
,
−
2
2
◦
11
π
6
√
300
√ 2
2
,
−
2
2
√
−
θ=
=
=
√ 3
1
,
−
2
2
330 ◦
θ
√
x
(cos θ)
θ = 360◦ = 2π
5π
210
√ 4
0
,
2
2
θ = 0◦ = 0
θ = 180◦ = π
θ=
−
6
s=1
◦
30
radiu
√ 4
0
,
2
2
(0, 0)
− 21
=
◦
√
−
√ 3
1
2 , 2
√
=
◦
5
2π
3
13
θ=
=
=
=
1
2
√ 2
2
,
2
2
√
θ=
◦
θ
√ √ − 23 , 21
1
√ 1
3
,
2
2
√
π
2
√
√ 2
2
,
2
2
120
−
θ=
√ 1
3
,
2
2
√
−
θ = 90◦ =
1
60 ◦
1
1
2
45 ◦
=
−1
− 12
1
1
2
√
√
√
√
√
The coordinates in the picture above are written to show a pattern: 20 , 21 , 22 , 23 , 24 . This pattern
appears in both the x-coordinates and the y-coordinates (but the pattern goes one direction for the
x-coordinates
and the
opposite direction
for the y-coordinates). Some of these numbers are usually
√
√
√
simplified: 20 = 0, 21 = 21 , and 24 = 1.
Right-triangle trigonometry (for 0 < θ <
π
2)
hyp
opp
θ
adj
sin θ =
opp
hyp
cos θ =
adj
hyp
tan θ =
opp
adj
csc θ =
hyp
opp
sec θ =
hyp
adj
cot θ =
adj
opp
Pythagorean theorem: In a right triangle with legs a and b and hypotenuse c,
a2 + b2 = c2 .
c
a
b
Trigonometric identities
tan θ =
sin θ
cos θ
cot θ =
1
cos θ
=
tan θ
sin θ
sec θ =
1
cos θ
csc θ =
1
sin θ
sin2 θ + cos2 θ = 1, for every value of θ
cos(−θ) = cos θ
sin(−θ) = − sin θ
)
for every value of θ
There are many more trigonometric identities, but the ones above are the most important. See Chapter 15
of the textbook, “Trigonometric Identities,” for more identities, including the angle addition formulas.