Precalculus Solutions | Expert Assistance for Problems & Examples

Maximum value of ${(3 \cos (x - 0.730) - 3)}^{2}$
Maximum value of $(3 \cos (x - 0.730)$ is 3 and minimum is -3
Conceptually , to find the maximum of ${(3 \cos (x - 0.730) - 3)}^{2}$ is to use ${(3 - 3)}^{2}$ however that is 0.But when I use the minimum value ${(- 3 - 3)}^{2}) = 36$ and 36 is the maximum value . But isn't this conceptually wrong to use the minimum value to find the maximum value ?
I do not understand why must we use -3 obviously squaring a negative number will give u a positive number . But is there a reason why must I use -3 ?

Breanna Mcclure 2022-04-14

Least Positive integer satisfying the condition $x^{2} - 4 x > \cot^{- 1} x$

Jazmin Strong 2022-04-14

Is there a function satisfying the following equation: $f (\sin x) + f (\cos x) = \frac{\tan x}{2}$

Destinee Bryan 2022-04-14

How do I go about solving this derivative of inverse tangent?
$f (x) = 8 \tan^{- 1} (\frac{y}{x}) - \ln (\sqrt{x^{2} + y^{2}})$

Cazzaoro0w9 2022-04-14

For $0 < x < \frac{π}{4}$ prove that $\frac{\cos x}{\sin^{2} x (\cos x - \sin x)} > 8$

Willie Kelley 2022-04-14

Find the minimum value of the expression
$y = \frac{16 - 8 \sin^{2} 2 x + 8 \cos^{4} x}{\sin^{2} 2 x}$

Cash Duncan 2022-04-14

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 $4 \csc^{2} x + 9 \sin^{2} x$

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