Solving Trigonometry Equations and Problems with Examples and Tips

Find the limit $lim_{x \to \frac{π}{2}} (\frac{\cos (5 x)}{\cos (3 x)})$ without using L'Hospital's rule

Kasey Castillo 2022-04-14

Find the inverse function of $\tan^{2} (x) - 2 \sqrt{3} \tan (x)$
The question is pretty clear, we must find the inverse function of
$f : x \to \tan^{2} (x) - 2 \sqrt{3} (\tan (x))$
This function is a bijection from $(- \frac{π}{2} + k π, \frac{π}{3} + k π)$ to $(- 3, + \infty)$ since it is strictly decreasing and continuous.
I've tried factorizing with tangent and playing with the equation so I can add Arctangent to eliminate tangent like with usual functions but I do not seem to be capable succeeding using this method.
P.S: This function is also a bijection from $(- \frac{π}{3} + k π, \frac{π}{2} + k π)$ to $(- 3, + \infty)$

Jazmin Strong 2022-04-14

Find the derivative:
$y = \arcsin (2 x + 1)$

Perla Benitez 2022-04-14

Find $\tan (x + y)$ given $\sin x = 3 \sin (x + 2 y)$
My reference gives the solution $- 2 \tan y$ but how do I prove it ?
My Attempt
$3 = \frac{\sin (x + 2 y)}{\sin x} \Rightarrow 3 \sin x = \sin x . \cos 2 y + \cos x . \sin 2 y$
$3 \sin x = 2 \sin x . \cos^{2} y - \sin x + 2 \cos x . \sin y . \cos y = 2 \cos y . \sin (x + y) - \sin x$
$\Rightarrow 4 \sin x = 2 \cos y . \sin (x + y) \Rightarrow \sin (x + y) = 2 . \frac{\sin x}{\cos y}$
$\cos (x + y) = \frac{\sin x}{\sin y} \Rightarrow \tan (x + y) = 2 \tan y$

Ariel Collier 2022-04-14

Find minimum value of $8 \cos x + 4 \sin x$ and corresponding value of x
I used the R method to simplify it -
$\sqrt{80} \cos (x - 26.565)$
Minimum value of that =
$\sqrt{80} \cos (x - 26.565) = - \sqrt{80}$
$\cos (x - 26.565) = - 1$
This cosine value lies in the 2nd and 3rd quadrant
letting x-26.565=y
y reference angle = $\cos^{- 1} (- 1) = 180$
2nd quadrant - 180-y(ref)=0
3rd quadrant - 180+y(ref)=360
Therefore , x=26.565,386.565
Why am I wrong ? The minimum value is 206.6

Blaine Jimenez 2022-04-14

Find all x in the interval $(0, \frac{π}{2})$ such that
$\frac{\sqrt{3} - 1}{\sin x} + \frac{\sqrt{3} + 1}{\cos x} = 4 \sqrt{2}$

Emilia Hoffman 2022-04-14

Do the equation
$2 \cos^{2} (\frac{x}{2}) \sin^{2} (\frac{x}{2}) = x^{2} + \frac{1}{x^{2}}$

Rosa Townsend 2022-04-13

Maximizing $3 \sin^{2} x + 8 \sin x \cos x + 9 \cos^{2} x$ . What went wrong?
Let $f (x) = 3 \sin^{2} x + 8 \sin x \cos x + 9 \cos^{2} x$ . For some $x \in [0, \frac{π}{2}], f$ attains its maximum value, m. Compute $m + 100 \cos^{2} x$
What I did was rewrite the equation as $f (x) = 6 \cos^{2} x + 8 \sin x \cos x + 3$ . Then I let $a = < 6 \cos x, 8 \cos x > and b = < \cos x, \sin x >$ .
Using Cauchy-Schwarz, I got that the maximum occurs when $\tan x = \frac{4}{3}$ and that the maximum value is $10 \cos x$ . However, that produces a maximum of 9 for f(x), instead of the actual answer of 11.

Axel Stout 2022-04-13

Maximizing the sum $\sum_{n = 1}^{m} \sin n$
For which value of m, we will obtain the maximum sum?
Here's my approach : $\sum_{n = 1}^{m} \sin n = \frac{\sin 1}{4 \sin^{2} \frac{1}{2}} - \frac{2 \cos (m + \frac{1}{2})}{4 \sin (\frac{1}{2})}$
If we can minimize $2 \cos (m + \frac{1}{2})$ then this will result in maximizing the sum.
But the problem is I can't quite figure out what the minimum value of $\cos (m + \frac{1}{2})$ is.

Lillianna Sandoval 2022-04-13

Knowing that $α$ is acute and that $\sin α - \cos α = \frac{1}{3}$ , determine the exact value of
$\tan^{2} α - \cot^{2} α$
I've used $\tan α = \frac{\sin α}{\cos α}$ but can't seem to get anywhere. I get stuck here:
$\frac{\sin α + \cos α}{3 \cos^{2} α \sin^{2} α}$
when i use $\sin^{2} α + \cos^{2} α = 1$ here it seems to get even more complicated.

fanairana7lu1 2022-04-13

Is $x = \arccos (\frac{3}{5})$ a rational number?

Joaquin Salas 2022-04-13

Is $\cot x = \tan (\frac{π}{2} - x)$ true for any angle x?

Delilah Novak 2022-04-13

Find the number of real solutions on an interval $(0, π)$ of this equation
$\sin (14 u) - \sin (12 u) + 8 \sin (u) - \cos (13 u) = 4$

Briley Cabrera 2022-04-13

Find the least value of $4 \csc^{2} x + 9 \sin^{2} x$

Delilah Novak 2022-04-13

Find the limit and prove the limit for
$lim_{(x, y, z) \to (1, 2, - 3)} \arctan (\frac{x + z}{y})$

Axel Stout 2022-04-13

Find the general solution of
My approach: Converted them to cosines and then further simplified. I got or But my book stated the answer as or Where am i going wrong.
My simplification:
$\cos 2 θ - \cos 4 θ = 2 \cos 4 θ \cdot \cos 2 θ$
$\cos 2 θ - \cos 4 θ = \cos 6 θ + \cos 2 θ$
$\cos 4 θ = - \cos 6 θ$
$\cos 4 θ = \cos (π - 6 θ)$
$4 θ = 2 n π \pm (π - 6 θ)$
taking the plus and minus, I got my above-stated answer. But it doesn't quite agree with the answer given in my book. Can anyone explain my mistake?

Perla Benitez 2022-04-13

Find the calue without using calculator
$\sin^{2} 10 + \cos^{2} 40 + \sin 10 \cos 40$

Willow Cooper 2022-04-13

Find the domain of $\frac{1}{\cos^{2} (θ) + \cos (θ)}$

Eddie Clarke 2022-04-13

Find $\sum_{n = 1}^{89} \sin^{6} (n) = \frac{m}{n}$

Elijah Schwartz 2022-04-13

find $\cos θ$ if $\sin θ = \frac{3}{4}$ and $\tan θ = \frac{9}{2}$

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