Solving Trigonometry Equations and Problems with Examples and Tips

How to solve
$3 \sin^{2} x - 5 \sin x \cos x + 2 \cos^{2} x = 0$

Dexter Odom 2022-04-01

Range of $y = \cot^{- 1} x \cot^{- 1} (- x)$
Attempt: Let $\cot^{- 1} x = θ$ . Then $y = θ (π - θ)$ ;since $θ \in (0, π)$ .Then $θ^{2} - π θ + y = 0$ .For $θ$ to be real, $y \leq \frac{π^{2}}{4}$
Thus $0 < y \leq \frac{π^{2}}{4}$

Dumaen80p3 2022-04-01

Question: Realise the denominator $\frac{3 + 2 i \sin θ}{1 - 2 i \sin θ}$ and hence find θ if the expression is purely imaginary.
I've realised the denominator $\frac{3 + 8 i \sin θ - 4 \sin^{2} θ}{1 + 4 \sin^{2} θ}$ but don't know how to utilise the knowledge that the expression is purely imaginary.

tambustqaze 2022-04-01

$i \sin (x)$ phase shift
By Euler's formula, I can express i in the following way:
$i = \cos (\frac{π}{2}) + i \sin (\frac{π}{2}) = \exp (i \frac{π}{2})$
I wonder if it is legitimate to write
$i \sin x = \exp (i \frac{π}{2}) \cdot \frac{\exp (i x) - \exp (- i x)}{2 i}$
$= \frac{\exp (i x) \exp (i \frac{π}{2}) - \exp (- i x) \exp (i \frac{π}{2})}{2 i}$
$= \frac{\exp (i (x + \frac{π}{2})) - \exp (- i (x + \frac{π}{2}))}{2 i}$
$= \sin (x + \frac{π}{2})$
I don't feel like this is right, because it would imply a lot of weird things. So where is my mistake?

Alexis Alexander 2022-04-01

Prove that $(1 + 2 \cos \frac{2 x}{3}) \sin \frac{x}{3} = \sin x$ for $x \in R$

markush35q 2022-04-01

Prove that $\frac{\cos 3 A + \cos 3 B}{2 \cos (A - B) - 1} = (\cos A + \cos B) \cos (A + B) - (\sin A + \sin B) \sin (A + B)$
I used $\cos 3 A = 4 \cos^{3} A - 3 \cos A$ , but it is getting more and more complicated.

Sanai Huerta 2022-04-01

Prove that $\frac{\sec 8 A - 1}{\sec 4 A - 1} = \frac{\tan 8 A}{\tan 2 A}$
I tried converting $\sec 4 A$ into $\frac{1}{\cos 4 A}$ and $\sec 8 A$ into $\frac{1}{\cos 8 A}$ but couldn't get the answer. Any help would be appreciated.

Asher Olsen 2022-04-01

Prove that
$16 \cos^{5} A - 20 \cos^{3} A + 5 \cos A = \cos 5 A$

Adolfo Hebert 2022-04-01

Prove that
$2 \sin^{- 1} \sqrt{x} - \sin^{- 1} (2 x - 1) = \frac{π}{2}$

Beryneingmk39 2022-04-01

Prove $4 \sin (2 \frac{π}{5}) + \tan (2 \frac{π}{5}) = 5 \cot (\frac{π}{5})$
Attempt: Let the side length be 1. Then by breaking down the pentagon into 5 congruent triangles, it is easy to show that the area is $A = \frac{5}{4} \cot (\frac{π}{5}) = \frac{5}{2} R^{2} \sin (2 \frac{π}{5}) = 5 r^{2} \tan (\frac{π}{5})$ where R is the circumradius and r is the inradius.
Then it suffices to prove that the area can also be written as $\sin (2 \frac{π}{5}) + \frac{1}{4} \tan (2 \frac{π}{5})$ I tried to break down the area of a regular pentagon as a trapezoid and triangle, but the algebra got messy: $A = (1 + \cos (2 \frac{π}{5})) \sin (2 \frac{π}{5}) + (\frac{1}{2}) (1 + 2 \cos (2 \frac{π}{5})) (R + r - \sin (2 \frac{π}{5}))$

Rowan Callahan 2022-04-01

Prove that
$(\sqrt{3} - 4 \sin (\frac{2 π}{15})) \cos (\frac{π}{30}) = \sin (\frac{π}{30})$

jncuenodd4nf 2022-04-01

Prove $\sec^{2} \frac{π}{7} + \sec^{2} \frac{2 π}{7} + \sec^{2} \frac{3 π}{7} = 24$ by using the roots of the polynomial $x^{3} - 21 x^{2} + 35 x - 7 = 0$

kembdumatxf 2022-04-01

Prove ${\tan A}_{1} < \frac{\sum_{i = 1}^{n} {\sin A}_{1}}{\sum_{i = 1}^{n} {\cos A}_{1}} < {\tan A}_{n}$ for $0 < A_{1} < A_{2} < \dots < A_{n} < \frac{π}{2}$

Asher Olsen 2022-04-01

Prove identity: $\sin α = \frac{2 \tan \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}}$
I am having a problem proving this identity. I write tan like $\frac{\sin \frac{α}{2}}{\cos \frac{α}{2}}$ and the squared one in the same way. I eventually get
$\frac{2 \sin \frac{α}{2} \cos \frac{α}{2}}{\sin^{2} \frac{α}{2} + \cos^{2} \frac{α}{2}}$

Guadalupe Glass 2022-03-31

The equation is
$\sqrt{3 + 4 \cos^{2} (x)} = \frac{\sin (x)}{\sqrt{3}} + 3 \cos (x)$

Milton Robertson 2022-03-31

The correct way of simplifying $\frac{4 \cos^{2} 2 α - 4 \cos^{2} α + 3 \sin^{2} α}{4 \cos^{2} (\frac{5 π}{2} - α) - \sin^{2} 2 (α - π)}$
Using half angle formula, I wrote it as follows
$\frac{4 \cos^{2} 2 α - 4 \cos^{2} α + 3 \sin^{2} α}{4 \cos^{2} (\frac{5 π}{2} - α) - \sin^{2} 2 (α - π)} = \frac{4 \cos 4 α - 7 \cos 2 α + 3}{2 - 2 \cos 2 α - \frac{1 - \cos 4 α}{2}}$
But I can't figure out what to do next. In fact, I tried other ways of expressing it, but again had numbers that couldn't be simplified.

parksinta8rkv 2022-03-31

I need help showing that $(4 \sin θ) (\sin (θ - \frac{π}{3})) (\sin (θ - \frac{2 π}{3})) = \sin (3 θ)$

Beryneingmk39 2022-03-31

I have tried to evaluate
$\int \frac{\sin (8 x)}{9 + \sin^{4} (4 x)} d x$

sempteim245 2022-03-31

I have to simplify and evaluate this :
${\cos 70}^{\circ} + 4 {\cos 70}^{\circ}$

palmantkf4u 2022-03-31

I have to show that
$\sin (\frac{π}{3}) = \frac{1}{2} \sqrt{3}$

The majority of Trigonometry Math problems are quite easy to solve when you have the list of answers that help you see the logic and reasoning. Looking at various examples will help you to see how certain questions are related to provided solutions.

Remember that Trigonometry requires patience and analysis. By following your analysis skills, you will help yourself find the answers. If something remains unclear, start with your exploration of examples again and go step-by-step.

Precalculus

Practice Exercises for Trigonometry