Precalculus Solutions | Expert Assistance for Problems & Examples

Can you help me prove: $\frac{y}{\sqrt{y^{2} + x}} = arccot \frac{y}{\sqrt{x}}$ ?

Damaris Huffman 2022-03-18

Can I simplify $n \cdot r \cdot \sin (90^{\circ} - \frac{180^{\circ}}{n}) \sqrt{r^{2} - r^{2} \sin^{2} (90^{\circ} - \frac{180^{\circ}}{n})}$ further?

pinka1hf 2022-03-18

Calculate $\cos (\frac{2 π}{5})$ in terms of $\sin (\frac{π}{5})$
I intend to calculate $\cos (\frac{2 π}{5})$ via the formula
$\sin (\frac{π}{5}) = \sqrt{\frac{1 - \cos (\frac{2 π}{5})}{2}}$
This could be expressed as:
$\sqrt{\frac{5 - \sqrt{5}}{8}} = \sqrt{\frac{1 - \cos (\frac{2 π}{5})}{2}}$ . Squaring both sides and manipulate a bit, this will give $\cos (\frac{2 π}{5}) = - \frac{1 - \sqrt{5}}{4}$ . However, the true result is $\frac{\sqrt{5} - 1}{4}$ . What is wrong with my derivation?

fellesturduj 2022-03-17

Alternative form of $y = \frac{1}{1 - {(\frac{x}{a})}^{2}}$ ?
I solved the differential equation $y^{'} (x) = \frac{2}{a^{2}} x y^{2} (x)$ (with y(0)=1 and $x \in (- | a |, | a |)$ ) and came to
$y (x) = \frac{1}{1 - {(\frac{x}{a})}^{2}} .$
This looks like some trigonometric thing to me. Do you know a better form for y?

Charity Barr 2022-03-17

Alternative proof of
$\cos^{6} θ + \sin^{6} θ = \frac{1}{8} {(5 + 3 \cos 4 θ)$

ohmaig00dnessyb3 2022-03-17

A question on range of trigonometric functions.
If $u = {\sqrt{a^{2} \cos^{2} α + b^{2} \sin^{2} α}} + {\sqrt{a^{2} \sin^{2} α + b^{2} \cos^{2} α}}$ , find the difference between the maximum and minimum value of $u^{2}$
I tried squaring the expression on both sides but i am ending up with some complex expression, i.e, $a^{2} + b^{2} + 2 \sin α \cos α \sqrt{{(a^{2} - b^{2})}^{2} + \frac{a^{2} b^{2}}{\sin^{2} α \cos^{2} α}}$

evighetl2b 2022-03-17

What is the maximum value of k so that $x^{2} + 4 (\sin^{2} {x} \tan^{2} {x} + \cos^{2} {x} \cot^{2} {x} + k^{2} - x \sec {x} \csc {x}) = 0$ has real roots.

voordragh2 2022-03-17

Choose $y \in R$ so that $\sin y = \frac{u}{\sqrt{u^{2} + v^{2}}}$ and $\cos y = \frac{v}{\sqrt{u^{2} + v^{2}}}$
I know the identity $\sin^{2} y + \cos^{2} y = 1$ . Also, I notice that
${(\frac{u}{\sqrt{u^{2} + v^{2}}})}^{2} + {(\frac{v}{\sqrt{u^{2} + v^{2}}})}^{2} = 1$
without u,v vanishing simultaneously. But I am not sure whether $\exists y \in R$ s.t. $\sin y = \frac{u}{\sqrt{u^{2} + v^{2}}}$ and $\cos y = \frac{v}{\sqrt{u^{2} + v^{2}}}$ .Does anyone have an idea?

WesDiectstemiwxg 2022-03-17

Check the proof that $f (x) = \cos ({x}) \cdot \cos ({\sqrt{3} x})$ is not periodic
If the function is periodic then:
$f (x) = f (x + T)$
$\cos (x) \cdot \cos (\sqrt{3} x) = \cos (x + T) \cdot \cos (\sqrt{3} (x + T))$
Consider the function at 0:
$\cos (T) \cdot \cos (\sqrt{3} T) = 1$
But this equation has only one solution at T=0 which contradicts the initial assumption that there exists a positive period.
Or the other way:
$\cos (T) \cdot \cos (\sqrt{3} T) = 1$
Let $\cos (T) = 1 and \cos (\sqrt{3} T) = 1$ , hence
$T = 2 π m$
$\sqrt{3} T = 2 π n$
Substituting T in the second equation gives:
$2 π m \sqrt{3} = 2 π n$
$\sqrt{3} = \frac{n}{m}$
But $m, n \in N$ and $\sqrt{3} \in {R ∖ Q}$ which gives a contradiction.

afadimoz4 2022-03-17

Calculate:
$\tan (α + 15^{\circ}) \cdot {\cot 165}^{\circ}$ , if
${\sin 75}^{\circ} \cdot \sin (α + 15^{\circ}) = 2 \sin α$

drulluna60h 2022-03-17

Calculate $lim_{x \to 0^{+}} \frac{\arctan (\log (1 + \sqrt{x})) \sin^{3} (x^{\frac{3}{4}})}{(e^{\tan (x)} - 1) (1 - \sin^{2} (x))}$

lindamonkey2ix 2022-03-17

Bijectivity of sinx on the interval $[- π, π]$ to $[- 1, 1]$
how can I prove that sinx is a bijective map from the domain $[- π, π]$ to the co-domain $[- 1, 1]$ . I had no problem proving that using the graphical representation of sinx, but rigorously could not.

chechemuaen7 2022-03-16

An inequality with $\cos$ and triangle sides
Here is the problem:
Let ABC be a triangle with sides a,b,c. Show that $\frac{\cos A}{a^{3}} + \frac{\cos B}{b^{3}} + \frac{\cos C}{c^{3}} \geq \frac{3}{2 a b c}$

e3r6a2n1dz60 2022-03-16

A trigonometric equation ${(\sin {\frac{π}{7}})}^{x} + {(\cos {\frac{π}{7}})}^{x} = 1$

Anika Wolfe 2022-03-16

A trigonemetry problem
If $\sec (θ + α) \sec (θ - α) = 2 \sec (θ)$
and $\cos (θ) = K \cos \frac{α}{2}$
Then prove that $K = \pm \sqrt{2}$
I have converted the $\sec$ into $\cos$ and then applied the formulaes of $\cos (θ + α) \cos (θ + α)$ we get
$2 \sin^{2} α = 2 \cos^{2} θ - \cos θ$
Then $2 \cos^{2} (θ) = K^{2} (1 + \cos α)$ from the other equation
But i can't get K by solving both the equations

Kaleb Riddle 2022-03-16

What is the precise value of the expression
$\frac{{\cos 25}^{\circ} + \sin (- 5^{\circ})}{{\sin 25}^{\circ} + {\cos 5}^{\circ}}$

Riley Quinn 2022-03-16

Can someone prove that $\sum_{i = 1}^{89} \sin^{2 n} (\frac{π}{180} i)$ is a dyadic rational for all positive integers n, or find counterexample?

Gavyn Trujillo 2022-03-16

Can someone please help me with this question:
$\int \frac{16}{1 - \cos 8 x} d x$

Kienastsrx 2022-03-16

Calculate ${\tan 9}^{\circ} - {\tan 27}^{\circ} - {\tan 63}^{\circ} + {\tan 81}^{\circ}$ ?

Payton Benson 2022-03-15

What is the value of this trigonometric sum ratio:
$\frac{\sum_{n = 1}^{44} {\cos n}^{\circ}}{\sum_{n = 1}^{44} {\sin n}^{\circ}} = ?$

Precalculus problems and solutions are hard to find online even though this topic is quite popular among college students majoring in Engineering. Keeping this fact in mind, we decided to provide you with a list of questions and answers that will help you with various precalculus problems. Regardless if it is related to equations or any other problem, these precalculus problems and composite functions examples and solutions below will help you see the logic and compare solutions as you check your original task.

Precalculus

Solve Precalculus Problems | Expert Help Available