In the diagram below 𝐾(− 6; 𝑦) is a point such that 𝑂𝐾 = 10 units and reflex 𝑋𝑂𝐾 = θ.

Question

Figure 3: Diagram for question 3.
3.1
Calculate the value of 𝑦.
(2)
3.2
Determine the value of each of the following (without using a calculator):
3.2.1
𝑠𝑖𝑛(180° − θ)
(2)
3.2.2
𝑐𝑜𝑠(180° + θ)
(2)
3.3
𝐿isapointon𝑂𝐾suchthat𝑂𝐿 = 5units.
Determine the coordinates of 𝐿 (without using a calculator).
(4)
[10]

Answer 1

3.1
Since OK = 10 units and XKO is a right angle, we can use the Pythagorean theorem to find the value of y.

OK^2 = OX^2 + XK^2
10^2 = (6)^2 + y^2
100 = 36 + y^2
y^2 = 64
y = ±8

Since K is in the second quadrant, y = -8.

3.2.1
sin(180° - θ) = sin(180°)cos(θ) - cos(180°)sin(θ) = 0 * cos(θ) - (-1) * sin(θ) = sin(θ)

3.2.2
cos(180° + θ) = cos(180°)cos(θ) - sin(180°)sin(θ) = -1 * cos(θ) - 0 * sin(θ) = -cos(θ)

3.3
Since OX = 6 units and OL = 5 units, the x-coordinate of L is 6 - 5 = 1. Since K is in the second quadrant, the coordinates of L are (1, -8).