UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
PREPARING FOR UNIVERSITY MATH PROGRAM. PUMP-2017-WINTER.
TRIGONOMETRY DEFINITIONS
1. Unit Circle: The unit circle is the circle of radius 1 centred at the origin in the 𝑥𝑦-plane with the
equation: 𝑥 ! + 𝑦 ! = 1.
2. Angle in standard position: An angle in standard position is an angle drawn in the 𝑥𝑦-plane
with its vertex at the origin and its initial side on the positive 𝑥-axis. See examples below.
Recall that if the rotation is counterclockwise, the angle is considered positive, and if the rotation
is clockwise, the angle is considered negative.
3. Radian measure: Consider the unit circle. The radian measure of an angle in standard position
is the length of the arc (on the unit circle) that is associated with the angle. Note: If the angle is
positive or negative, then so is the radian measure. Also note that 360° = 2𝜋 rad.
4. Terminal point determined by 𝜽: The terminal point determined by 𝜃 is the point 𝑃(𝑥, 𝑦) on
the unit circle that we land on after travelling a distance of 𝜃 radians from the point (1,0).
5. Reference angle (R.A.): A reference angle is a radian measure associated with one of the
following special angles: 30°, 45°, and 60°. Reference angles are used to describe the shortest
standard distance along the unit circle, between a point inside of a quadrant and the 𝑥-axis
(as shown below).
𝜋
𝜋
𝜋
30° = rad,
45° = rad,
60° = rad
6
4
3
6. Cosine function: The cosine function is the function 𝑓, defined by 𝑓 𝜃 = cos 𝜃, where cos 𝜃
is the 𝑥-coordinate of the terminal point 𝑃(𝑥, 𝑦) determined by the radian 𝜃. Note that
𝑑𝑜𝑚 𝑓 = ℝ and 𝑟𝑎𝑛(𝑓) = [−1, 1].
7. Sine function: The sine function is the function 𝑓, defined by 𝑓 𝜃 = sin 𝜃, where sin 𝜃 is
the 𝒚-coordinate of the terminal point 𝑃(𝑥, 𝑦) determined by the radian 𝜃. Note that
𝑑𝑜𝑚 𝑓 = ℝ and 𝑟𝑎𝑛(𝑓) = [−1, 1].
!"# !
8. Tangent function: The tangent function is the function 𝑓, defined by 𝑓 𝜃 = tan 𝜃 = !"# !,
such that 𝑑𝑜𝑚 𝑓 =
!!!! !
!
| 𝑘 ∈ ℤ} and 𝑟𝑎𝑛(𝑓) = ℝ, .
9. Periodic function: A function 𝑓 is periodic if there exists a 𝑝 ∈ ℝ! such that 𝑓 𝑥 + 𝑝 = 𝑓(𝑥)
for every 𝑥 ∈ 𝑋. Note: 𝑝 is called the period of 𝑓.
10. Even function: A function 𝑓 is even if 𝑓 −𝑥 = 𝑓(𝑥) for every 𝑥 ∈ 𝑋. The graph of an even
function is symmetric about the 𝑦-axis.
11. Odd function: A function 𝑓 is odd if 𝑓 −𝑥 = −𝑓(𝑥) for every 𝑥 ∈ 𝑋. The graph of an odd
function is symmetric about the origin.
12. Relative inverse cosine function: The relative inverse cosine function is the function, written
𝑓 𝑥 = arccos 𝑥 or 𝑓 𝑥 = cos !! 𝑥, with range 𝑌 = [0, 𝜋].
It is defined by:
arccos 𝑥 = 𝑦
⇔
cos 𝑦 = 𝑥
13. Relative inverse sine function: The relative inverse sine function is the function, written
𝑓 𝑥 = arcsin 𝑥 or 𝑓 𝑥 = sin!! 𝑥, with range 𝑌 = [−𝜋/2, 𝜋/2].
It is defined by:
arcsin 𝑥 = 𝑦
⇔
sin 𝑦 = 𝑥
14. Relative inverse tangent function: The relative inverse tangent function is the function,
written 𝑓 𝑥 = arctan 𝑥 or 𝑓 𝑥 = tan!! 𝑥, with range 𝑌 = (−𝜋/2, 𝜋/2).
It is defined by:
arctan 𝑥 = 𝑦
⇔
tan 𝑦 = 𝑥