Trigonometric expressions
Not what you're looking for?
1) 2 tan (theta /2)=
2) 2 tan theta Sin ^2 (theta /2)=
3) 2 Cos (45 + x) Cos (45-x)=
4) (Sin 3 x - 3 Sin x )/ ( Cos 3 x + 3 Cos x) =
5) 2 tan (theta /2) / { 1 + tan ^2 (theta /2) } =
6) Find the least positive value of theta such that
tan (45+ theta ) - 3 tan theta =2
Purchase this Solution
Solution Summary
Simplifies trigonometric expressions are examined.
Solution Preview
Note:
^ denotes power and * denotes multiplication
1) 2 tan (theta /2)=
From trigonometric identities
tan (alpha +beta ) = (tan alpha + tan beta ) / ( 1- tan alpha tan beta )
Putting alpha =beta =theta /2 in the identity
tan (theta /2+theta /2) = (tan theta /2 + tan theta /2) / ( 1- tan theta /2 tan theta /2)
or tan theta = 2 tan theta /2 / (1- tan ^2 theta /2)
or 2 tan theta /2 = tan theta * (1- tan ^2 theta /2)
Alternatively:
tan theta /2 = Sin theta /2 / Cos theta /2
Multiplying both the numerator and denominator of RHS by Cos theta /2
tan theta /2 = (Sin theta /2 Cos theta /2) / (Cos theta /2 Cos theta /2)
Multiplying both sides by 2
2 tan theta /2 = (2 Sin theta /2 Cos theta /2) / (Cos theta /2 Cos theta /2)
But Sin theta = 2 Sin theta /2 Cos theta /2
Cos theta = Cos (theta /2 + theta /2 ) = Cos ^2 theta /2 - Sin ^2 theta /2
or 1 + Cos theta = 1 + Cos ^2 theta /2 - Sin ^2 theta /2
But 1- Sin^2 theta /2 = Cos ^2 theta /2
Therefore 1 + Cos theta = 2 Cos ^2 theta /2
or Cos ^2 theta = (1 + Cos theta )/2
Substituting Sin theta for 2 Sin theta /2 Cos theta /2
and (1 + Cos theta )/2 =Cos ^2 theta
We get
2 tan (theta /2) = Sin theta / ( 1+ Cos theta ) /2
or
2 tan (theta /2) = 2 Sin theta / ( 1+ Cos theta )
Answer: 2 tan (theta /2) = 2 Sin theta / ( 1+ Cos theta )
2) 2 tan theta Sin ^2 (theta /2)=
Cos theta = Cos^2 (theta /2) - Sin ^2 (theta /2) (Identity)
Subtracting both left hand side (LHS) and right hand side( RHS) from 1
or 1- Cos theta = 1- {Cos^2 (theta /2) - Sin ^2 (theta ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Probability Quiz
Some questions on probability