Precalculus Solutions | Expert Assistance for Problems & Examples

Prove that
$2 \sin^{- 1} \sqrt{x} - \sin^{- 1} (2 x - 1) = \frac{π}{2}$

Asher Olsen 2022-04-01

Prove that
$16 \cos^{5} A - 20 \cos^{3} A + 5 \cos A = \cos 5 A$

Rowan Callahan 2022-04-01

Prove that
$(\sqrt{3} - 4 \sin (\frac{2 π}{15})) \cos (\frac{π}{30}) = \sin (\frac{π}{30})$

Beryneingmk39 2022-04-01

Prove $4 \sin (2 \frac{π}{5}) + \tan (2 \frac{π}{5}) = 5 \cot (\frac{π}{5})$
Attempt: Let the side length be 1. Then by breaking down the pentagon into 5 congruent triangles, it is easy to show that the area is $A = \frac{5}{4} \cot (\frac{π}{5}) = \frac{5}{2} R^{2} \sin (2 \frac{π}{5}) = 5 r^{2} \tan (\frac{π}{5})$ where R is the circumradius and r is the inradius.
Then it suffices to prove that the area can also be written as $\sin (2 \frac{π}{5}) + \frac{1}{4} \tan (2 \frac{π}{5})$ I tried to break down the area of a regular pentagon as a trapezoid and triangle, but the algebra got messy: $A = (1 + \cos (2 \frac{π}{5})) \sin (2 \frac{π}{5}) + (\frac{1}{2}) (1 + 2 \cos (2 \frac{π}{5})) (R + r - \sin (2 \frac{π}{5}))$

kembdumatxf 2022-04-01

Prove ${\tan A}_{1} < \frac{\sum_{i = 1}^{n} {\sin A}_{1}}{\sum_{i = 1}^{n} {\cos A}_{1}} < {\tan A}_{n}$ for $0 < A_{1} < A_{2} < \dots < A_{n} < \frac{π}{2}$

jncuenodd4nf 2022-04-01

Prove $\sec^{2} \frac{π}{7} + \sec^{2} \frac{2 π}{7} + \sec^{2} \frac{3 π}{7} = 24$ by using the roots of the polynomial $x^{3} - 21 x^{2} + 35 x - 7 = 0$

Asher Olsen 2022-04-01

Prove identity: $\sin α = \frac{2 \tan \frac{α}{2}}{1 + \tan^{2} \frac{α}{2}}$
I am having a problem proving this identity. I write tan like $\frac{\sin \frac{α}{2}}{\cos \frac{α}{2}}$ and the squared one in the same way. I eventually get
$\frac{2 \sin \frac{α}{2} \cos \frac{α}{2}}{\sin^{2} \frac{α}{2} + \cos^{2} \frac{α}{2}}$

find a formula for the quadratic function whose graph has the given y-intercept of y=-1 and a vertex at (-1,-2)

Guadalupe Glass 2022-03-31

The equation is
$\sqrt{3 + 4 \cos^{2} (x)} = \frac{\sin (x)}{\sqrt{3}} + 3 \cos (x)$

Milton Robertson 2022-03-31

The correct way of simplifying $\frac{4 \cos^{2} 2 α - 4 \cos^{2} α + 3 \sin^{2} α}{4 \cos^{2} (\frac{5 π}{2} - α) - \sin^{2} 2 (α - π)}$
Using half angle formula, I wrote it as follows
$\frac{4 \cos^{2} 2 α - 4 \cos^{2} α + 3 \sin^{2} α}{4 \cos^{2} (\frac{5 π}{2} - α) - \sin^{2} 2 (α - π)} = \frac{4 \cos 4 α - 7 \cos 2 α + 3}{2 - 2 \cos 2 α - \frac{1 - \cos 4 α}{2}}$
But I can't figure out what to do next. In fact, I tried other ways of expressing it, but again had numbers that couldn't be simplified.

parksinta8rkv 2022-03-31

I need help showing that $(4 \sin θ) (\sin (θ - \frac{π}{3})) (\sin (θ - \frac{2 π}{3})) = \sin (3 θ)$

Beryneingmk39 2022-03-31

I have tried to evaluate
$\int \frac{\sin (8 x)}{9 + \sin^{4} (4 x)} d x$

encaderpwp 2022-03-31

I have to simplify:
$\frac{(x - 2)}{(3 x + 6)} \cdot \frac{(x^{2} - x - 6)}{(x + 3)}$

sempteim245 2022-03-31

I have to simplify and evaluate this :
${\cos 70}^{\circ} + 4 {\cos 70}^{\circ}$

palmantkf4u 2022-03-31

I have to show that
$\sin (\frac{π}{3}) = \frac{1}{2} \sqrt{3}$

r1fa8dy5 2022-03-31

Suppose we are given a function
$g (x) = \sum_{n = 1} \frac{\sin (n x)}{10^{n} \sin (x)}, x \neq k π, k \in Z$

Nathanael Hansen 2022-03-31

I have to prove that
$\prod_{k = 1}^{n - 1} (1 - cis (\frac{2 k π}{n})) = n$

Nancy Richmond 2022-03-31

I have this problem:
$9 x^{3} - 18 x^{2} - 4 x + 8 = 0$

tambustqaze 2022-03-31

Sum of $1 - {\cos θ}_{i}$ is bounded
For each $i = 1, \dots, N$ let $θ_{i} \geq 0$ be angles such that $\sum_{i = 1}^{N} θ_{i} \leq π$ . Prove that $\sum_{i = 1}^{N} (1 - {\cos θ}_{i}) \leq 2$

svrstanojpkqx 2022-03-31

Summation of $\sum_{r = 1}^{n} \frac{\cos (2 r x)}{\sin ((2 r + 1) x \sin ((2 r - 1) x}}$
My Try:
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x) \sin ((2 r + 1) x - (2 r - 1) x)}{\sin 2 x : \sin ((2 r + 1) x \sin ((2 r - 1) x}}$
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x) (\sin ((2 r + 1) x \cos (2 r - 1) x - \cos (2 r - 1) x) \sin (2 r + 1) x)}{\sin 2 x \sin ((2 r + 1) x \sin ((2 r - 1) x)}}$
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x)}{\sin 2 x} (\cot (2 r - 1) x - \cot (2 r + 1) x)$

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.

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