Solving Trigonometry Equations and Problems with Examples and Tips

Prove that $\sin^{4} θ + \cos^{4} θ = \frac{3 + \cos 4 θ}{4}$
By considering $(\sin^{2} θ + \cos^{2} θ)^{2}$ and $(\sin^{2} θ + \cos^{2} θ)^{3}$ prove that
$a) \sin^{4} θ + \cos^{4} θ = \frac{3 + \cos 4 θ}{4}, b) \sin^{6} θ + \cos^{6} θ = \frac{5 + 3 \cos 4 θ}{8}$

Petrovcic2x 2022-06-24

How do i find the exact value of
$\arccos (\frac{\sqrt{2}}{\sqrt{3}})$
Any ideas on how to find the exact value of the expression without using a calculator?

varitero5w 2022-06-24

How to prove $\frac{\sin (π + α)}{\sin (\frac{3 π}{2} + α)} + \frac{\cos (α - π)}{\cos (\frac{π}{2} + α) + 1} = \frac{1}{\cos α}$ ?

Jeramiah Campos 2022-06-24

Prove that $\frac{\sec (16 A) - 1}{\sec (8 A) - 1} = \frac{\tan (16 A)}{\tan (4 A)}$

Winigefx 2022-06-24

Finding value of $\prod_{k = 1}^{n - 1} \cos \frac{k π}{2 n} = \frac{\sqrt{n}}{2^{n - 1}}$
Try: let
$z^{2 n} = 1 \Rightarrow z = (1)^{\frac{1}{2 n}} = e^{\frac{k π}{n}}$
So
$z^{2 n} - 1 = \prod_{k = 1}^{2 n} (z - e^{\frac{k π}{n}}) = (z^{2} - 1) \prod_{k = 1}^{n - 1} (z - e^{\frac{k π}{n}}) \cdot (z - e^{\frac{- k π}{n}})$
So
$z^{2 n} - 1 = (z^{2} - 1) \prod_{k = 1}^{n - 1} (z^{2} - 2 z \cos (k π / n) + 1$
Could some help me how to prove my original formula

minwaardekn 2022-06-24

If $\frac{1}{a + b} + \frac{1}{b + c} = \frac{3}{a + b + c}$ , prove that $∠ B = 60^{\circ}$
My Attempt
$\frac{1}{a + b} + \frac{1}{b + c} = \frac{3}{a + b + c}$
$\frac{a + 2 b + c}{(a + b) (b + c)} = \frac{3}{a + b + c}$
$a^{2} - a c - b^{2} + c^{2} = 0$
How to prove further?

landdenaw 2022-06-24

If $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac{1}{2}$ , then find $\frac{\sin 2 x}{\sin 2 y} + \frac{\cos 2 x}{\cos 2 y}$

Yesenia Sherman 2022-06-24

How to simplify $ℜ [\sqrt{2} \tan^{- 1} \frac{x}{\sqrt{i}}]$ ?

Winigefx 2022-06-24

Find the interval for which $2 \arctan x + \arcsin \frac{2 x}{1 + x^{2}}$ is an independent of x?

Adriana Ayers 2022-06-24

Simplify $\cot (2 α) - \cot (α)$ without using the double angle formula.
I tried to solve it by the following way:
$\frac{1}{\tan (α + α)} - \frac{1}{\tan (α)} = \frac{- (1 + \tan^{2} (α))}{2 \tan (α)} = \frac{2 \tan (α)}{\cos^{2} (α)} .$

Emanuel Keith 2022-06-24

Prove that $\sin^{6} \frac{θ}{2} + \cos^{6} \frac{θ}{2} = \frac{1}{4} (1 + 3 \cos^{2} θ)$

anginih86 2022-06-24

I'm trying to find all intervals $[a, b]$ on which the functions $\sin (2 π t)$ and $\cos (2 p i t)$ are orthogonal.
$\int_{a}^{b} \sin (2 π t) \cdot \cos (2 π t) d t = \frac{\cos (4 π b) - c o s (4 π a)}{8 π} = 0$
$⟺ \cos (4 π b + 2 π k) = c o s (4 π a + 2 π l), k, l \in Z$
I don't know how to solve this for a and b, can anybody help me with that please?

Sarai Davenport 2022-06-24

How to get to the other forms of the identity $\cos 2 x = \cos^{2} x - \sin^{2} x$

Lydia Carey 2022-06-24

Solve: $\cos^{2} x > \sin^{2} x$

Erin Lozano 2022-06-24

Solve the following equation: $\begin{aligned} \sin^{14} x + \cos^{14} x = \frac{169}{64} \cos^{6} 2 x \end{aligned}$

opepayflarpws 2022-06-24

Find all z such that $| \tan z | = 1$

Jackson Duncan 2022-06-24

Evaluate:
$\csc^{2} (\frac{π}{9}) + \csc^{2} (\frac{2 π}{9}) + \csc^{2} (\frac{4 π}{9}) = ???$

Winigefx 2022-06-24

trigonometry equation - $\sin^{3} (x) + \sin^{3} (2 x) + \sin^{3} (3 x) = (\sin (x) + \sin (2 x) + \sin (3 x))^{3}$

Kyla Ayers 2022-06-24

Prove that
$1 + (\binom{n}{1}) \cos x + (\binom{n}{2}) \cos 2 x + . . . \cos n x = (2 \cos \frac{x}{2})^{n} (\cos \frac{n x}{2})$
given that
$(1 + \cos x + i \sin x)^{n} = (2 \cos \frac{x}{2})^{n} (\cos \frac{n x}{2} + i \sin \frac{n x}{2})$
I'm not so sure how to even start with this, any hints?

Jeffery Clements 2022-06-24

How to find the parametric equation of: $x^{2} - 3 y^{2} = 1$

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