Your question is insufficiently precise, but I'll try to answer anyway.
"Sine squared theta" usually means "the value of the sine of theta, quantity squared".
"Sine theta squared" usually means "the value of the sine of the quantity theta*theta".
The two are not at all the same.
It is 2*sin(theta)*sin(theta) because that is how multiplication is defined!
2 sin^2 theta = 1/4 sin^2 theta = 1/8 sin theta = sqrt(1/8) theta = arcsin(sqrt(1/8))
If there is a plus in between, that would be equal to 1, as a result of the Pythagorean Theorem. Otherwise, you can convert this into other forms with some of the trigonometric identities for multiplication, but you won't really get it into a simpler form.
It's 1/2 of sin(2 theta) .
If r-squared = theta then r = ±sqrt(theta)
It is 2*sin(theta)*sin(theta) because that is how multiplication is defined!
1
Remember use the Pythagorean Trig/ Identity. Sin^(2)(Theta) + Cos^(2)(Theta) = 1 Algebraically rearrange Sin^(2)(Theta) = 1 - Cos^(2)(Theta) Substitute Sin^(2)(Theta) = 1 - 0.65^(2) Factor Sin^(2)(Theta) = ( 1- 0.65 )( 1 + 0.65) Sin^(2)(Theta) = (0.35)(1.65) Sin^(2)(Theta) = 0.5775 Sin(Theta) = sqrt(0.5775) Sin(Theta) = 0.759934207.... Theta = Sun^(-1)(0.759934207...) Theta = 49.45839813 degrees.
Until an "equals" sign shows up somewhere in the expression, there's nothing to prove.
-Sin^(2)(Theta) + Cos^(2)Theta => Cos^(2)Theta - Sin^(2)Theta Factor (Cos(Theta) - Sin(Theta))( Cos(Theta) + Sin(Theta)) #Is the Pythagorean factors . Or -Sin^(2)Theta = -(1 - Cos^(2)Theta) = Cos(2)Theta - 1 Substitute Cos^(2)Thetqa - 1 + Cos^(2) Theta = 2Cos^(2)Theta - 1
2 sin^2 theta = 1/4 sin^2 theta = 1/8 sin theta = sqrt(1/8) theta = arcsin(sqrt(1/8))
COS squared Theta + SIN squared Theta = 1; where Theta is the angles measurement in degrees.
Cosine squared theta = 1 + Sine squared theta
The expression (\cos^2(90^\circ - \theta)) can be simplified using the co-function identity, which states that (\cos(90^\circ - \theta) = \sin(\theta)). Therefore, (\cos^2(90^\circ - \theta) = \sin^2(\theta)). This means that (\cos^2(90^\circ - \theta)) is equal to the square of the sine of (\theta).
sin(0)=0 and sin(very large number) is approximately equal to that same very large number.
Tan^2
The expression "cot theta = 1.5 sin theta" can be rewritten using the definitions of trigonometric functions. Since cotangent is the reciprocal of tangent, we have cot(theta) = cos(theta) / sin(theta). Therefore, the equation becomes cos(theta) / sin(theta) = 1.5 sin(theta), leading to cos(theta) = 1.5 sin^2(theta). This relationship can be used to find specific values of theta that satisfy the equation.