Solving Trigonometry Problems: Basic, Advanced & Examples

High School
Geometry
Trigonometry

Using De Moivre's law to compute ${(- \sqrt{3} + i)}^{\frac{2}{3}}$
Question:
If $z = - \sqrt{3} + i$ , then $z^{\frac{2}{3}} = ?$

stropa0u 2022-01-24

If $y = \arctan (\frac{2 x - 1}{1 + x - x^{2}})$ , then $\frac{d y}{d x}$ at x=1 is equal to?
I have tried 2 different approaches, both yielding different results and both results are present in the options. Am I doing something wrong or why is this happening?

Miguel Davenport 2022-01-24

If a,b,c are positives such that $a + b + c = \frac{π}{2}$ and $\cot (a), \cot (b), \cot (c)$ is in arithmetic progression, find $\cot (a) \cot (c)$

Jazmin Perry 2022-01-24

If $g (x) = f (\tan^{2} x - 2 \tan x + 4), 0 < x < \frac{π}{2}$ , then g(x) is increasing in what interval?
I have first differentiated the given equation and found the value of x for which the interior part of f is 3 ,this gives $x = \frac{π}{4}$ and it gives me $g (\frac{π}{4}) = 0$ but I am stuck here and don't know what to do after this.

Maximus George 2022-01-24

Find the maximum angle X in the range $0^{\circ} \leq x \leq 360^{\circ}$ which satisfies the equation $\cos^{2} (2 x) + \sqrt{3} \sin (2 x) - \frac{74}{=} 0$

Tori Hines 2022-01-24

Why must x be acute for the identity $\sqrt{\frac{1 - \sin x}{1 + \sin x}} = \sec x - \tan x$ to hold true?

poveli1e 2022-01-24

If $\cos 2 v = - \frac{19}{}$ and v is acute, then determine the value of v
I have tried using the double angle identity
$\cos 2 v = \cos^{2} v - \sin^{2} v$
Ive

Adrien Barron 2022-01-23

Determine the greatest possible value of
$\sum_{i = 1}^{10} \cos 3 x_{i}$
for real numbers $x_{1}, x_{2} \dots x_{10}$ satisfying
$\sum_{i = 0}^{10} {\cos x}_{i} = 0$
My attempt:
$\sum \cos 3 x = \sum 4 \cos^{3} x - \sum 3 \cos x = 4 \sum \cos^{3} x$
So now we have to maximize sum of cubes of ten numbers when their sum is zero and each lie in interval [−1,1]. i often use AM GM inequalities but here are 10 numbers and they are not even positive. Need help to how to visualize and approach these kinds of questions.

Sereinserenormg 2022-01-23

The range of $f (x) = \cos^{2 n} (x) + \sin^{2 n} (x)$
I have $f (x) = \cos^{2 n} (x) + \sin^{2 n} (x), n \in N, n \geq 2, x \in R$ I need to find the range of the function. I took n=2 and I got $f (x) = 1 - {\frac{12}{\sin}}^{2} (2 x)$ and the range of this is [1/2,1]
Also, for n=3 I got [1/4,1].
How to find the range for n?

Jaylene Franco 2022-01-23

Finding solutions to $3 \tan^{2} θ - \tan θ - 14 = 0$ within $0 \leq θ \leq 360^{\circ}$

Jamya Elliott 2022-01-23

Solving $4 \sin^{4} x - 4 \sin^{2} x + 1 = 0, x \in [0, 2 π ⟩$
I am having problem solving this trig equation. Every time I try to simplify or substitute sinx with u, I get $x = \emptyset$ .
How do I go about solving this equation?
$4 \sin^{4} x - 4 \sin^{2} x + 1 = 0, x \in [0, 2 π ⟩$

Jamarion Kerr 2022-01-23

Solve the trigonometry equation in rational form.
$\frac{\tan^{2} x \sin^{2} x}{1 - \sin^{2} x \cos 2 x} + \frac{\cot^{2} x \cos^{2} x}{1 - \cos^{2} x \cos 2 x} + \frac{2 \sin^{2} x}{\tan^{2} x + \cot^{2} x} = \frac{32}{}$
What I thought:
$\frac{\tan^{2} x \sin^{2} x}{1 - \sin^{2} x \cos 2 x} + \frac{\cot^{2} x \cos^{2} x}{1 - \cos^{2} x \cos 2 x} + \frac{2 \sin^{2} x}{\tan^{2} x + \cot^{2} x} - \frac{32}{=} 0$
Transform each fraction into $\tan x$ , then you can factor out ${(\tan x - 1)}^{2} {(\tan x + 1)}^{2}$

Kendall Holder 2022-01-23

P(t) models the distance of a swinging pendulum (In CM) from the place it has travelled t seconds after it starts to swing. Here, t is entered in radians.
$P (t) = - 5 \cos (2 π t) + 5$
What is the first time the pendulum reaches 3.5 CM from the place it was released?
Round your final answer to the hundredth of a second.
Ok, so I did this to get the solution:
$3.5 = - 5 \cos (2 π t) + 5$
$- 1.5 = - 5 \cos (2 π t)$
$0.3 = \cos (2 π t)$
$\cos - 1 (0.3) = 2 π t$
$\frac{\cos^{- 1} (0.3)}{2 π} = t$
The fact is this gives me 11.55 seconds to get to 3.5 CM, which does not sound right. Where did I go wrong and how can I fix it?

Lainey Goodwin 2022-01-23

How to prove the identity $\sin 2 x + \sin 2 y = 2 \sin (x + y) \cos (x - y)$

poveli1e 2022-01-23

How can you calculate $\arctan (2) + \arctan (3)$ 's value?

bistandsq 2022-01-23

How do you evaluate $\sec [\cot^{- 1} (- 6)]$ ?

Nataly Best 2022-01-23

How do you find $\sin (\sin^{- 1} (\frac{1}{4}))$ ?

meteraiqn 2022-01-22

How do you find the $\arcsin (\sin (\frac{7 π}{6}))$ ?

Fallbasiss4 2022-01-21

I have a non-right-angled isosceles triangle with two longer sides, X, and a short base Y.
I know the length of the long sides, X.
I also know the acute, vertex angle opposite the base Y, lets

Carly Shannon 2022-01-21

What is the inradius of the octahedron with sidelength a?
1) Determine the height of one of two constituent square pyramids by considering a right triangle, using the fact that the height of the equilateral triangle is $a \sqrt{\frac{3}{2}}$ . (Incidentally, by this you obtain the circumradius of the Octahedron.)
2) Now cut the (solid) octahedron along 4 of these latter heights, i.e. across two non-adjacent vertices and two midpoints of parallel sides of the ''square base''. (Figure 1)
Consider the resultant rhombic polygon, whose sidelength is $a \sqrt{\frac{3}{2}}$ . Choose a convenient pair of adjacent corners A and B, s.t. the angle in A is obtuse. Then the insphere touches AB, say in M.
So the question becomes: Is there a still simpler proof?

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Trigonometry

Trigonometry Problems - Basics, Advanced & Examples