CHAPTER REVIEW TEST 2
B. PRIMARY DIRECTED ANGLES
1. Coterminal Angles
Definition standard pposition oof aan aangle
An angle in the coordinate plane whose vertex is at the origin and whose initial side lies along the positive x-axis is said to be in standard pposition.
Definition coterminal aangles
Two or more angles whose terminal sides coincide with each other when they are in standard position are called coterminal angles.
Let us look at an example of coterminal angles. The figure shows a unit circle.
The positive angle ∠AOP corresponds to the arc AùEP and the negative angle ∠AOP corresponds to the arc AùFP. These angles are coterminal. The measure of the positive angle ∠AOP is m(∠AOP) = α°
and the measure of the negative angle ∠AOP is m(∠AOP) = –(360 – α)°.
We can also express the measure of each angle in radians. Since this is a unit circle, if the length of the arc AùEP is θ then the measure of the positive angle ∠AOP is m(∠AOP) = θ and the measure of the negative angle ∠AOP is m(∠AOP) = –(2π – θ).
Now assume that point P in the figure is moving around the circumference of the unit circle from point A in the counterclockwise direction. Study the following table.
O P
E
A q a
F
y
x
Position oof ppoint P (moving ccounterclockwise) Measure oof tthe ccentral aangle ffor AïP
Degrees Radians
0° 0
P lies on the positive x-axis P lies on the positive y-axis 90°
180° π
P lies on the negative x-axis P lies on the negative y-axis 270°
360° 2π
P lies on the positive x-axis after one complete revolution
720° + α 4π + θ
P lies on the ray [OP after a second revolution
k⋅ 360° + α 2kπ + θ
P lies on the ray [OP after its kthrevolution
360° + α 2π + θ
P lies on the ray [OP after one complete revolution
0π
3π 2
Solution
EXAMPLE
3
For each angle, write the set of coterminal angles with the same unit of measurement.a. 175° b. 5
4 π
Coterminal angles differ by an integral multiple of complete angles.
a. {175° + k⋅ 360°, k ∈ ]} = {...,–545°, –185°, 175°, 535°, 895°,...}
b. {5 + 2 , }= {..., 11 , , , , ,...}3 5 13 21 4π k⋅ π k∈] − 4π − 4π 4π 4π 4π
EXAMPLE
4
Find the arc length which corresponds to the central angle 40° on the unit circle (π≅ 3).Solution Since we have We know that on the unit circle, the radian measure of a directed angle is equal to the length of the directed arc corresponding to the angle. So the arc length is and using π ≅ 3 gives us 2 2 3 2 0.6.
9 9 3
π ≅ ⋅ = ≅ 2 ,
9 π
40 = , so =2 .
180 9
R R
° π
= π 180
D R
π
Note
The angles in part a and part b are coterminal. Therefore, if we graph them in standard position, these angles will have the same terminal side.
185°
175°
3p 4
5p 4
a. b.
Solution
EXAMPLE
5
Find the coordinates of the terminal point of the arc with length which is in standardposition on the unit circle. 2
π
The circumference of a unit circle measures 2π.
So represents a quarter of the circle.
Therefore the arc length corresponds to the point B(0, 1) on the unit circle.
Furthermore, the arc length π corresponds to the point A′(–1, 0) on the unit circle and
corresponds to B′(0, –1).
3 2
π 2
π 2
π
A(1, 0) A¢(1, 0)
B¢(0, 1) B(0, 1) y
x
2. Primary Directed Angles and Arcs
In order to find a primary directed angle α we must divide the initial angle by 360. We must not simplify before the division, because 360 represents a complete rotation. For example, the remainder in the operation 5000÷360 gives us the required primary directed angle whereas the simplified version 500÷36 does not.
In other words, the primary directed angle of β is the smallest positive angle that is coterminal with β. If we divide β by 360°, the remainder will be the primary directed angle.
m(β ) = k ⋅ (360°) + m(α), k ∈ ].
For example, 30° is the primary directed angle of 390° because 390° = 1 ⋅ 360° + 30°.
We know that the radian measure of any angle is equal to the length of the arc which corresponds to its central angle in the unit circle. The circumference of a unit circle is 2π.
Therefore any two real numbers that differ by integral multiples of 2π will coincide at the same point on the circle.
Definition primary ddirected aangle
Let β be an angle which is greater than 360°. Then the positive angle α ∈ [0, 360°) which is coterminal with β is called the primary ddirected aangle of β.
Definition primary ddirected aarc
The positive real number t∈ [0, 2π) which differs from a real number by integral multiples of 2π is called a primary ddirected aarc.
Since t is the smallest positive real number that is coterminal with a given angle θ, we can find t by subtracting integral multiples of complete rotations from θ, or alternatively by dividing θ by 2π and considering the remainder:
θ = k ⋅ (2π) + t, k ∈ ].
EXAMPLE
6
Find the primary directed angle of each angle, using the same unit.a. 7320° b. –7320° c. d. 75
8
− π 75
8 π
Solution a. 7320 360 7200
120
– 20
number of rotations
7320° = (20 ⋅ 360°) + 120°, so 120° is coterminal with 7320°.
So the primary directed angle of 7320° is 120°.
20 complete rotations in the positive direction y
x
21st rotation 120°
y
x
b. Solution 11
7320° = (20 ⋅ 360°) + 120°
–7320° = –(20 ⋅ 360° + 120°)
= (–20 ⋅ 360°) – 120°
= –120°
– 120° ≡ 240° (coterminal angles) – 7320° ≡ 240° (coterminal angles)
Solution 22
–7320° = (–21) ⋅ 360° + 240°
Therefore the primary directed angle of –7320° is 240°.
20 complete rotations in the negative direction y
x
21st rotation +240°
120°
y
x
c. Solution 11
Solution 22
1. Divide the numerator by twice the denominator:
2 ⋅ 8 = 16 and 75 ÷ 16 = (4 ⋅ 16) + 11.
2. Multiply the remainder by : 11 11 . denominator 8 8
π ⋅ =π π
number of rotations 75 =64 +11
d. If the angle was positive, the remainder would be as we found in part c. But the angle is negative, so the remainder is Because a coterminal angle must be positive, we calculate
So the primary directed angle is 5 . 8
number of rotations remainder
EXAMPLE
7
Find the primary directed angle of θ = −30° 42′ 15′′.Solution The primary directed angle must be positive, so we need to find the positive difference from 360°. Let the primary directed angle be θ ′.
To make the calculation easier we can write 360° as 359° 59′ 60′′. Then 359° 59′ 60′′
Check Yourself 2
1. Find the primary directed angle of each angle, using the same unit of measurement.
a.100° b.7200° c. d.
e.–400° f.–50° g. h.
2. For each angle, write the set of coterminal angles with the same unit of measurement.
a.30° b.120° c. d.
Consider the right triangle in the figure. The table shows the trigonometric ratios for the acute angle θ.