Solving Trigonometry Equations and Problems with Examples and Tips

Find $lim_{n \to \infty} \cos (\frac{π}{4}) \cos (\frac{π}{8}) \dots \cos (\frac{π}{2^{n}})$

Jay Mckay 2022-01-26

How is the trig identity $\tan^{2} (x) = \sec^{2} (x) - 1$ derived?

Ethen Wong 2022-01-26

Solving $\frac{d y}{d θ} + y \cos θ = \frac{12}{\sin 2} θ$ with $y (\frac{π}{2}) = 4$

Fiona Petersen 2022-01-26

If $\cos^{4} x . \sec^{2} y, \frac{1}{2}, \sin^{4} x . \csc^{2} y$ are in A.P, then prove that $\cos^{8} x . \sec^{6} y, \frac{1}{2}, \sin^{8} x . \csc^{6} y$ in A.P
My Attempt
$\cos^{4} x . \sec^{2} y + \sin^{4} . x \csc^{2} y = \frac{\cos^{4} x}{\cos^{2} y} + \frac{\sin^{4} x}{\sin^{2} y} = 1$
$\Rightarrow \sin^{2} y . \cos^{4} x + \cos^{2} y . \sin^{4} x = \sin^{2} y . \cos^{2} y$
$\cos^{8} x . \sec^{6} y + \sin^{8} x . \csc^{6} y = \frac{\cos^{8} x}{\cos^{6} y} + \frac{\sin^{8} x}{\sin^{6} y}$
How do I know that the given terms are in A.P, G.P or H.P ?

mairie0708zl 2022-01-26

Proof that $| \arctan (x) - \frac{π}{4} - \frac{(x - 1)}{2} | \leq \frac{{(x - 1)}^{2}}{2}$
I'm trying to prove that, for every $x \geq 1$ :
$| \arctan (x) - \frac{π}{4} - \frac{(x - 1)}{2} | \leq \frac{{(x - 1)}^{2}}{2}$
I could do it graphically on $R$ , but how to make a formal algebraic proof?

Shamar Padilla 2022-01-26

The challenge trignometry question is: simplify $\sin (80^{\circ}) + \sin (40^{\circ})$ using trignometric identities. All the angle values are in degrees.
This is what I did: Let $a = 40^{\circ}$ . So we have $\sin (2 a) + \sin (a)$ . Using the formula for $\sin (2 a)$ , I have,
$2 \sin (a) \cos (a) + \sin (a) = \sin (40) (2 \cos (40^{\circ}) + 1)$
All I got was a big red mark through the question. I looked at co-functions and half-angles and couldn't find a solution that came out to something I could do without a calculator. Any thoughts?

Gavin Frye 2022-01-26

Values of x satisfying $\sin x \cdot \cos^{3} x > \sin^{3} x \cdot \cos x$
My Attempt
$\sin x \cos x \cdot (\cos^{2} x - \sin^{2} x) = \frac{1}{2} \cdot \sin 2 x \cdot \cos 2 x = \frac{1}{4} \cdot \sin 4 x > 0 ⟹ \sin 4 x > 0 x \in (0, π) ⟹ 4 x \in (0, 4 π)$
$4 x \in (0, π) \cup (2 π, 3 π) \Rightarrow x \in (0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4})$
But, my reference gives the solution, $x \in (0, \frac{π}{4}) \cup (\frac{3 π}{4}, π)$ where am I going wrong with my attempt?

Aaden Lam 2022-01-26

Given $\tan α = 2$ , evaluate $\frac{\sin^{3} α - 2 \cos^{3} α + 3 \cos α}{3 \sin α + 2 \cos α}$

siutulysr5 2022-01-26

Given $y = A \cos (k t) + B \sin (k t)$ where A,B,k are constants, prove that $y^{(2)} + (k^{2}) y = 0$
There was a silly typo, $y^{n}$ is actually y′′

Rudy Koch 2022-01-26

How to rewrite $\cos 2 x$ in terms of $\frac{1 - \sin x}{1 + \sin x}$
I had rewritten $\cos (2 x)$ in terms of $\tan (x)$
$\cos 2 x = \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$

minikim38 2022-01-26

Evaluating the integral $\int_{0}^{\frac{π}{4}} \sqrt{1 - 16 \sin^{2} x} d x$
I tried a substitution
$u = 4 \sin x, d x = \frac{d u}{\sqrt{16 - u^{2}}}$
hence the integration will be
$\int_{u = 0}^{u = 2 \sqrt{2}} \frac{\sqrt{1 - u^{2}}}{\sqrt{16 - u^{2}}} d u$
But I could not complete the solution using this substitution.

Jenny Branch 2022-01-26

How to solve the equation $2 z^{2} + i = - 2$ ?
Assuming that $k = - 1 - \frac{i}{2}$ we have $| k | = \frac{\sqrt{5}}{2}$ and
$\cos ϕ = \frac{- 2 \sqrt{5}}{5}, \sin ϕ = - \frac{\sqrt{5}}{5}$
How should I find $ϕ$ ?

Aubrey Hendricks 2022-01-26

Let $x \in R$ and let $n \in N$ then evaluate: $\sum_{k = 0}^{n} (\binom{n}{k}) \sin (x + \frac{k π}{2})$ I could only go up to breaking this sum in two parts of sinx and cosx in the following way: $\sum_{k = 0}^{n} (\binom{n}{k}) \sin (x + \frac{k π}{2}) = \sum_{k = 0}^{| n / 2 |} (- 1)^{k} (\binom{n}{2 k}) \sin x + \sum_{k = 0}^{| n / 2 |} (- 1)^{k} (\binom{n}{2 k + 1}) \cos x$ I have the intuition that $\sin (x + y) = \sin x \cos y + \sin y \cos x$ should be used at some point.

Emmy Combs 2022-01-26

Solving for x in $\sin^{- 1} (2 x) + \sin^{- 1} (3 x) = \frac{π}{4}$
I tried solving by differentiating both sides, but I get x=0.
How do you solve it, purely using trigonometric techniques?

Jessie Jenkins 2022-01-26

Integrating $\frac{1}{\sin (3 x)}$ when sovling an ODE

tripplek7i 2022-01-26

Prove the identity for $\tan 3 θ$
Using de Moivres

Jaeden Frederick 2022-01-25

I cannot find how or why this,
$5 \sin 3 x - 1 = 3$
Has one of two solutions being this,
$42.99 + n \times 120^{\circ}$
I am lost on how to get this solution. I have found the other one, so I will not mention it.

Journey Stout 2022-01-25

Proving $\arctan (\sqrt{x^{2} + 1}) = \frac{π}{4} - \frac{1}{2} \arctan (x)$

treetopssan 2022-01-25

Solve equation $1 + \sin^{2} θ = 3 \sin θ \cos θ (given {\tan 71}^{\circ} 34' = 3)$

Jazmin Perry 2022-01-25

Evaluate
$\int \frac{d x}{\sin x - \cos x}$
I know it can be done by Weierstrass substitution. But I am looking for new/simple approach. For example I tried:
$\int \frac{1}{\sin x - \cos x} \cdot \frac{\sin x + \cos x}{\sin x + \cos x} d x = \int \frac{\sin x + \cos x}{- \cos 2 x} d x$
but I can't continue from here.

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