Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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40 views

Is this a typo, or am I missing something?

I have a handout for my precalc II class. It says $\sinh(-x) = -\sin(x)$ It should be $\sinh(-x) = -\sinh(x)$ right? I don't see how a negative input could make a hyperbolic function circular.
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0answers
14 views

Solution to Tanθ = -3/4 in converting to cylindrical coordinates

I am attempting to convert (8, -6, 7) from rectangular coordinates into cylindrical coordinates. We have r = 10, but then I end up with tanθ = -3/4 and I am not sure how to get an exact answer for ...
4
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2answers
337 views

How can I prove this trigonometric statement true?

$$ {1+\sin^{2}\left(x\right) \over \cos^{2}\left(x\right)} = 1 + 2\tan^{2}\left(x\right)$$ This statement is part of a larger problem, but I need to prove that this is true before moving on. I'm ...
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1answer
30 views

Integral about lengh o f an arc

I can't find a way to solve this: $$ \int_{\pi/2}^{\pi} \sqrt{8sen^2(t)cos^2(t)}dt $$ The integral is to calculate the length of an arc, by parametric equations. The answer is $\sqrt{2}$, but i'm ...
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5answers
35 views

I have a trigonometry question that needs to be answered

Can someone please answer the question: If $\tan{2\alpha} = \frac{1}{2}$, find $\sin{2\alpha} $ and $\cos{2\alpha}$? Thank you.
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0answers
44 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...
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4answers
77 views

What is $\cos(k \pi)$?

I want to ask question for which I have been finding answer for. Please could anyone explain me why $\cos(k \pi) = (-1)^k$ and also explain me same for $\sin(k \pi)$?
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2answers
29 views

Sin & Cos Equation/Relation

If sin(x) = 0.3, find cos(pi-x) how i would solve this: let x = sin-1(0.3) solve for cos(pi-[sin-1(0.3)]) Is there a way to solve this by hand? Is the above method wrong?
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1answer
47 views

Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
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1answer
42 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
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0answers
39 views

An isosceles triangle with a measure of the sides abc of $5,5$ and $4$.

Find the angles of the triangle in an isosceles triangle of length 5 as the hypotenuse and $\sqrt{21}$ as the height of the triangle as well as the angle bisector, and measure the angles and find ...
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0answers
35 views

Polygones inscribed with in a circle

Let's say that there is a circle in two dimension and the diameter of the circle is 1.First start with an equilateral triangle inscribed with in the circle and the measure of the angles are equal to ...
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0answers
15 views

Trig Star problem

I need to find an angle where I have a radius of 50. The Radius starts at point A which Forms a 90 degree angle CAF the distance between C & F is 70.71. The angles For ACF and AFC are both 45 ...
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2answers
57 views

Finding the lengths of the sides of a triangle given 3 angles only.

If a right triangle ABC with an angle A at 90 degree, B 45 degree, C 45 degree is their a way of finding the length of the sides abc without knowing any of their lengths. Normally we use ...
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2answers
35 views

Unusual result when comparing trigonometry and Pythagoras in triangles.

I'm a Scottish Higher maths student. I was looking over some old textbooks, and came across a seemingly easy question, involving a circle within a triangle. I used the expected method to solve it; ...
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2answers
28 views

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$ I tried to factor and I got $2\cos^4(x)+(-2\sin^2(x)-3)(\cos^4(x)+\sin^4(x))$ but that doesn't lead me to my goal. ...
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1answer
21 views

Trigonometry, rewriting an expression

How do I rewrite $((\cos t)^3) - 2((\cos t)\cdot((\sin t)^2))$ to $3(\cos t)^3 - 2 \cos t$?
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0answers
23 views

Multiple Waves all in phase (Wave packets)

Suppose there are seven waves of slightly different wavelengths and amplitudes and we superimpose them (textbook is talking about wave packets). The wavelengths range from $\lambda _9 = 1/9$ to ...
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2answers
24 views

Trig Reduction With Pythagoras #2

If $\sin 10^\circ = p$, then determine $$\sin 280^\circ$$ in terms of $p$. $$\sin 280^\circ=\sin(180^\circ+100^\circ) = -\sin 100^\circ$$ $$-\sin 100^\circ = -\sin(90^\circ+10^\circ) = -\sin ...
1
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1answer
19 views

Trig Reduction with Pythagoras

If $\sin 10 = p$, then determine $$\tan^2 30^\circ \times \tan^2190^\circ$$ in terms of p.
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2answers
52 views

Trigonometry and Quadratic Equations

If $\tan x+\tan^2 x+\tan^3 x=1$ Then, find the value of $2\cos^6 x-2\cos^4 x+\cos^2 x$.
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1answer
35 views

Is the complex cosine function surjective?

Let $\cos z=\frac{e^{iz} - e^{-iz}}{2}$ be the complex cosine function. Then is $\cos:\mathbb{C}\rightarrow \mathbb{C}$ surjective? If so, how do i prove this?
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0answers
13 views

What is the domain of complex tangent function?

What is the domain of $\tan z =\frac{\sin z}{\cos z}$ ? Is the domain $D=\{z\in\mathbb{C} : \cos z \neq 0\}$? Or considering the Riemann sphere, is $\tan z$ defined on $D$ as $\infty$?
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2answers
65 views

Find the range of a function

How to find the range of the following trigonometric function $\sin^2x-5\sin x-6$. Can some one help me out. Thank you
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0answers
44 views

How and why is a trig equation different from a regular equation?

$\frac{\sin(x)}{\cot(x)}+\cos(x)=\sec(x)$ is a example for the question. When proving this identity work for all values for $x$, you will end up with either $\frac{1}{\cos(x)}=\frac{1}{\cos(x)}$ or ...
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3answers
344 views

An almost impossible limit [duplicate]

The following limit appeared in a qualification exam: Find the limit of $$\lim_{x \to 0} \left( \frac{\tan (\sin (x))-\sin (\tan (x))}{x^7} \right).$$ I ended up doing it in Mathematica, is there any ...
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5answers
96 views

Why $\cos(-\theta)$ gives positive values while in case of sine it is negative?

Why $\cos(-\theta)$ gives positive values while in case of sine it is negative? I mean $\cos(-\theta) = +\cos(\theta)$ $\sin(-\theta) = -\sin(\theta)$ $\tan(-\theta) = -\tan(\theta)$ and please ...
0
votes
1answer
53 views

Find the slope of intercecting line given angle

My question is very similar to this one, but I don't know how to modify the answer for an angle not relative to the origin. It's been way to long since math class. If I have the line that passes ...
3
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2answers
32 views

check my solution to indefinite integral problem with arccos

So we had homework it asked us to find $$\int\arccos(x)dx$$ I have found that $$\int\arccos(x)dx=x\arccos (x)+\sqrt{1-x^2}+c$$ Is this right?
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1answer
24 views

How can I show this trigonometric identity?

Using only the basic identities ($\sin^2{A}+\cos^2{A}=1$, $1+\cot^2 A=\csc^2{A}$ and $1+\tan^2 A=\sec^2{A}$) show that: $$ \frac{1}{\csc{A}-\cot{A}}-\frac{1}{\sin A}=\frac{1}{\sin A}-\frac{1}{\csc ...
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2answers
14 views

Finding the cosine of an acute angle given the obtuse cosin

I have the cosine of an obtuse angle and want to find the cosine of the acute angle: i.e. I have $\cos{(\theta_{1})}$ how can I find $\cos{(\theta_{2})}$?
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1answer
30 views

For what $n$ the set $\{\sin x, \cos x, (\sin x)^2, (\cos x)^2,…, (\sin x)^n, (\cos x)^n\}$ is linearly independent?

Under what condition of $n$ the following set $\{\sin x, \cos x, \sin^2x, \cos^2x,..., \sin^nx, \cos^n x\}$ is linearly independent? I tried to replace n=1,2,3 but I haven't get the general result. ...
0
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4answers
65 views

Is a trig function with a constant in its parenthesis a constant?

For example: Is $\sin(8)$ a constant? I want to know because my professor differentiated it to $0$ and that was the explanation he gave. Thanks
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2answers
35 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
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1answer
20 views

Find the magnitude of a parallelogram?

Ok so far I know that first I am supposed to find angle ABC and angle ACB using the law of sines sinA/a=SinB/b=SinC/c . So that is what I did I got $sinABC/159 = sin71.5/15$. I got ABC is 71.5 yet I ...
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1answer
36 views

Is this a valid trig identity for sin(a)sin(b)?

I'm looking at a solution for a problem and one of the steps says that: $\sin(100\pi t)\sin(500\pi t) = \frac{1}{2}[\sin(100\pi+500\pi)t-\sin(500\pi-100\pi)t]$ The thing is, I don't recognize that ...
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5answers
67 views

Trigonometric Limits - solution needed

how to solve this problem? (without using l'Hopital rule) $$\lim_{x\to π/2} \frac{1-\sin x+\cos x}{\sin 2x -\cos x}$$ thanks for helping.
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2answers
28 views

Transforming position function into a sinusoidal function?

Im trying to transform the position function $x(t)=3\sin(\pi t)+4\cos(\pi t)$, into a sinusoidal function of the form $x(t)=A\sin(\omega t+\phi )$. Im trying to follow the steps over at this ...
0
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1answer
19 views

Arccos and inequalities?

There is something I don't understand with arccos and inequalities. Suppose I have this inequality $cos(x) ≤ \frac{1}{2}$ Having $x = 90$, satisfies this since $cos(90) = 0$. Then since arccos is ...
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2answers
41 views

How to evaluate this integral using trigonometric substitution?

I am pretty sure that my answer is correct but given answer for the exercise from textbook Calculus James Steward was slightly different. Any idea to solve this: $$\int\frac{x}{\sqrt{x^2+x+1}} \, ...
3
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2answers
70 views

What type of triangle satisfies the equation $\cos(A)-\cos(B)+\sin(C)=0$?

A triangle with angle $A,B,C$ satisfies the equation $\cos(A)-\cos(B)+\sin(C)=0$. What type of triangle is this? Regular, acute, right, obtuse etc. I tried using sine and cosine rule, but no result. ...
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2answers
24 views

A simpler way to prove this trigonometric identity?

The question asks to prove: $$ \frac{\tan{A}}{1-\cot{A}}+ \frac{\cot{A}}{1-\tan{A}}=\sec{A}\csc{A}+1$$ using only: $$ \sin^2{A}+\cos^2{A}=1\;\; \text{ & }\; \;\tan^2{A}+1=\sec^2{A}\;\; \text{ ...
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3answers
45 views

have trouble with this limit question

a) By considering the areas of the triangle OAD, the sector OAC and the triangle OBC, show that $(\cos \theta)(\sin \theta) < \theta < \frac{\sin\theta}{\cos\theta}$ I find out: Area of ...
3
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2answers
143 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
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2answers
41 views

If $\cos A + \sec A = 3$, Then find the value of $\cos^3 A + \sec^3 A$. [closed]

If $\cos A + \sec A = 3$, then find the value of $\cos^3 A + \sec^3 A$. a) 9 b) 27 c) 18 d) 20
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1answer
19 views

From centroid and two vertexes to find the missing vertex of a triangle.

The centroid of triange $ABC$ is located at $P(14,14)$ with points $A(2,12)$ and $B(22,6)$ What are the coordinates of vertex $C$? Explain how you found the answer. I've gotten a question of the ...
3
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1answer
30 views

What is this trigonometric expression equal to?

If $\frac{cos{x}}{cos{y}}=\frac{a}{b}$, then $(a\times tan{x}+b\times tan{y})$ equals (A)$(a+b)cot{\frac{x+y}{2}}$ (B)$(a+b)tan\frac{x+y}{2}$ (C)$(a+b)(tan\frac{x}{2}+tan\frac{y}{2})$ ...
0
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3answers
33 views

Prove that the following is an identity

$\frac{\sin(x)}{\cot(x)} + \cos(x) = \sec(x)$ Note that this involves identities, so you can't treat it like an equation and multiply both sides by a number. ( when I multiplied both sides by ...
-1
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2answers
50 views

Trigonometric equation proof [closed]

We have that $\sin (A+B) = \cos (A-B)$. Prove that $A$ and/or $B$ must be $\displaystyle n\pi + \frac{\pi}{4}$, where $n$ is an integer.
2
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4answers
42 views

How to remember a particular class of trig identities.

Please how can I easily remember the following trig identities: $$ \sin(\pi-x)=\sin x \color{red}{\text{ and }} \cos(\pi-x)=-\cos x\\ \sin(\pi+x)=-\sin x \color{red}{\text{ and }} \cos(\pi+x)=-\cos ...