Solving Trigonometry Equations and Problems with Examples and Tips

If $\tan x = 3$ , then what is the value of $\frac{3 \cos 2 x - 2 \sin 2 x}{4 \sin 2 x + 5 \cos 2 x}$ ?

Keaton Good 2022-01-25

Can $\cos (36^{\circ})$ be calculated using known cosines like $\cos (37^{\circ})$ and the derivative formula
$f^{'} (x) \approx \frac{f (x + △ x) - f (x)}{△ x}$

Duncan Reed 2022-01-25

Finding the fundamental period of $- 6 \cos (5 π x)$
I am trying to find the fundamental period of $f (x) = - 6 \cos (5 π x)$ . I know a periodic function satisfies f(x)=f(x+p).
I know that $\cos (x)$ 's periodicity is $2 π$ as $\cos (x + 2 π) = \cos (x)$ so I just assumed that $- 6 \cos (5 π \cdot 1 + 2 π) = - 6 \cos (5 π)$ , which it is but apparently this is not correct for the fundamental period?

Lainey Goodwin 2022-01-25

How do I show ${(\csc θ - \cot θ)}^{2} \equiv \frac{1 - \cos θ}{1 + \cos θ}$ ?

Ahmed Stewart 2022-01-25

Proving $\frac{1 + \cos x}{2 + \sin x} < \frac{4}{3}$

maliaseth0 2022-01-25

Is the phase shift of $- 3 \sin (2 x + \frac{π}{2}) + 1$ equal to $- \frac{π}{4}$ or just $- \frac{π}{2}$ ?

FiessyFrimatsd0 2022-01-25

Prove that $\sum_{n = 1}^{\infty} \log \cos (\frac{1}{n})$ converges absolutely.

William Montgomery 2022-01-25

Find $\cot (\sum_{n = 1}^{23} \cot^{- 1} (1 + \sum_{k = 1}^{n} 2 k))$
I tried to simplify it but it became messy. How can I find $\sum_{n = 1}^{23} \cot^{- 1} (1 + n + n^{2}) ?$

Marquis Neal 2022-01-25

Can this equation be solved? $x + \sin (x) = \frac{11 π}{48}$

Kayleigh Dixon 2022-01-25

Graph of $7 \cos x + 5 \sin x = 2 k + 1$
The question is to find the number of integral values of k for which the equation
$7 \cos x + 5 \sin x = 2 k + 1$ has a solution.
I have solved it by converting the LHS into $\sqrt{74} \sin (x + α)$ , where $α = \arctan (\frac{7}{5})$ . And then making an inequality with RHS.

tripplek7i 2022-01-25

Determine the polar form of $\sin θ + i (1 + \cos θ)$

licencegpopc 2022-01-25

Find the value of the function $\frac{\cos (α - β)}{2}$
I solved it by squaring the two equations such that I get $\sin^{2} α + \sin^{2} β + 2 \sin α \sin β = \frac{441}{4225}$ And $\cos^{2} α + \cos^{β} + 2 \cos α \cos β = \frac{729}{4225}$
Adding them we get $2 + 2 (\sin α \sin β + \cos α \cos β) = \frac{1170}{4225}$
Then, $1 + \cos (α - β) = \frac{585}{4225}$
Therefore $\cos (α - β) = - \frac{3640}{4225}$
So now $\frac{\cos (α - β)}{2} = + or - \sqrt{\frac{585}{8450}}$

veceriraby 2022-01-25

If $\sin a + \sin c = 2 \sin b$ , show that $\tan \frac{a + b}{2} + \tan \frac{b + c}{2} = 2 \tan \frac{c + a}{2}$

Babenzgs 2022-01-25

Separate equations of the pair of lines $x^{2} + 2 x y \sec θ + y^{2} = 0$

Jamya Elliott 2022-01-25

If $a + b + c = 90^{\circ}$ , prove $\tan (a) \cdot \tan (b) + \tan (b) \cdot \tan (c) + \tan (c) \cdot \tan (a) = 1$

Aubrey Hendricks 2022-01-25

If $π < α < \frac{3 π}{2}$ then the expression $\sqrt{4 \sin^{4} α + \sin^{2} 2 α} + 4 \cos^{2} (\frac{π}{4} - \frac{α}{2})$

Marquis Neal 2022-01-25

What did I get wrong when solving $\int \frac{\sqrt{x^{2} - 1}}{x^{4}} d x$ ?
Let $x = \sec θ$
Then $d x = \sec θ \tan θ d θ$
$\int \frac{\sqrt{\sec^{2} θ - 1}}{\sec^{4} θ} \sec θ \tan θ d θ$
$= \int \frac{\sec θ}{\sec^{4} θ} \sqrt{\tan^{2} θ} \tan θ d θ$
$= \int \frac{1}{\sec^{3} θ} \tan^{2} θ d θ$
$= \int \frac{1}{\sec^{3} θ} \frac{\sec^{2} θ}{\csc^{2} θ} d θ$
$= \int \frac{1}{\sec θ} \frac{1}{\csc^{2} θ} d θ$
$= \int \cos θ \sin^{2} θ d θ$
Using u-substition, let $u = \sin θ$
Then $d u = \cos θ d θ and d x = \frac{1}{\cos θ} d u$
$\int \cos θ u^{2} \frac{1}{\cos θ} d u$
$= \int u^{2} d u$
$= \frac{u^{3}}{3} + C$
$= \frac{\sin^{3} θ}{3} + C$
Since $x = \sec θ, \sec^{- 1} (x) = θ$
$= \frac{\sin^{3} (\sec^{- 1} (x))}{3} + C$

Addison Gross 2022-01-25

Proof that $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1} \sin (n)}{n} = \frac{1}{2}$

Alvin Pugh 2022-01-25

Solving $\sin 3 θ = \frac{12}{}$ on the interval $[0, 2 π]$ . I don't understand where some solutions came from.

Laney Spears 2022-01-25

Solving $lim_{x \to 0} \frac{1}{x^{2}} - \cot^{2} (x)$
My approach :
$lim_{x \to 0} \frac{1}{x^{2}} - \frac{\cos^{2} x}{\sin^{2} x}$
$lim_{x \to 0} \frac{\sin^{2} (x) - x^{2} \cos^{2} (x)}{x^{2} \sin^{2} (x)}$

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