To solve the given equation, we will first factor the equation and use the Zero Product Property to solve for sinθ. Then, we will use the unit circle to find the exact values of θ that satisfy the equation.
Solving the Equation for sinθ
Let's start by factoring the expression on the left-hand side.
We obtained two values for sin θ. The sine of an angle in standard position is the y-coordinate of the point of intersection P of its terminal side and the unit circle.
P(x,y)=(cosθ,sinθ)
Since the unit circle has a radius of 1, no point that lies on it will ever have a y-coordinate of -5. Therefore, we can disregard one of the equations.
sin θ = -5 doesnothave a solution
To solve the equation sin θ =0, we need to consider the points on the unit circle that have a y-coordinate of 0.
We found two points that have 0 as the x-coordinate. Knowing that the measure of half a turn is 180^(∘), we can find the desired angle measures.
We found two solutions for the equation sin θ=0.
θ= 0^(∘) and θ= 180^(∘)
All Angles
Keep in mind that if we add or subtract a multiple of 360^(∘), the terminal side of the angle will be in the same position. This means that resulting angles will also be the solutions to the original equation. Therefore, we can now write all angles which are solutions to the original equation.
All Solutions
0^(∘)+ k * 360^(∘)
180^(∘)+ k * 360^(∘)
Where k is any integer. Let's try to simplify this, starting with 0^(∘)+ k * 360^(∘).
Notice that 2k is even. Therefore, the obtained expression is the same as the following.
even integer * 180^(∘)
Now, let's simplify 180^(∘)+ k * 360^(∘).
Notice that 2k+1 is not even. This time the obtained expression is the same as the following.
odd integer * 180^(∘)
We can write both obtained expressions as a single expression.
odd integer * 180^(∘) even integer * 180^(∘) ⇕ any integer* 180 ^(∘)
To make this expression even simpler we will call that integer n. Let's state all solutions to the original equation in a simplified form.
n* 180^(∘) ⇔ 0^(∘) + n* 180^(∘)
