1. Show that Sin3 Theta - Cos2 Theta = (1 - Sin Theta)(4Sin^2 Theta + 2Sin Theta - 1).
Without using a calculator, show that Theta = 18 degrees is an exact solution of the equation Sin3 Theta = Cos2 Theta.
Justifying your answer, find the exact values of:
i) Sin18 degrees.
ii) Sin234 degrees.
To prove that Sin3 Theta - Cos2 Theta = (1 - Sin Theta)(4Sin^2 Theta + 2Sin Theta - 1), we can start by expanding Sin3 Theta using the trigonometric identity:
Sin3 Theta = Sin(2 Theta + Theta)
= Sin2 Theta*Cos Theta + Cos2 Theta*Sin Theta
Now let's expand the right-hand side:
(1 - Sin Theta)(4Sin^2 Theta + 2Sin Theta - 1)
= (1 - Sin Theta)(4Sin^2 Theta + 2Sin Theta) - (1 - Sin Theta)
= 4Sin^2 Theta - 4Sin^3 Theta + 2Sin Theta - 2Sin^2 Theta + Sin Theta - 1 + Sin Theta - 1
= -4Sin^3 Theta + 2Sin Theta - 2Sin^2 Theta + 2Sin Theta - 2
= -4Sin^3 Theta - 2Sin^2 Theta + 4Sin Theta - 2
Now, let's simplify (-4Sin^3 Theta - 2Sin^2 Theta + 4Sin Theta - 2):
= -2(Sin^3 Theta + 4Sin^2 Theta - 2Sin Theta + 1)
= -2(Sin^2 Theta)(Sin Theta - 1) - 2(Sin Theta - 1)
= (Sin Theta - 1)(2Sin^2 Theta - 2)
As we can see, this is equal to Sin3 Theta - Cos2 Theta, which confirms the given identity:
Sin3 Theta - Cos2 Theta = (1 - Sin Theta)(4Sin^2 Theta + 2Sin Theta - 1)
Now, to show that Theta = 18 degrees is an exact solution of the equation Sin3 Theta = Cos2 Theta, we substitute Theta = 18 degrees into the equation and evaluate it without using a calculator:
Sin(3 * 18) = Cos(2 * 18)
Sin(54) = Cos(36)
To find the exact values of Sin18 degrees and Sin234 degrees, we can use the following trigonometric identities:
1) Sin(3Theta) = 3Sin(Theta) - 4Sin^3(Theta)
2) Sin(180 - Theta) = Sin(Theta)
3) Sin(360 + Theta) = Sin(Theta)
i) Sin18 degrees:
Using identity 1, let's write Sin(54) as Sin(3 * 18):
Sin(3 * 18) = 3Sin(18) - 4Sin^3(18)
Let's denote Sin18 as x for simplicity:
x = 3x - 4x^3
0 = 2x - 4x^3
0 = 2x(1 - 2x^2)
x = 0 (this is one solution, but it doesn't satisfy the equation), or x = ±(1/√2)
Since we're dealing with Sin, which is positive in the first and second quadrants, the value of Sin18 degrees must be (1/√2).
ii) Sin234 degrees:
Using identity 3, let's write Sin(234) as Sin(360 + 234):
Sin(360 + 234) = Sin(234)
Sin(234) = Sin(180 + 54)
Using identity 2, Sin(180 + 54) = Sin(54)
Since we have already found that Sin(54) = (1/√2), we can conclude that Sin234 degrees is also equal to (1/√2).
So the exact values of:
i) Sin18 degrees = (1/√2)
ii) Sin234 degrees = (1/√2)