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

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2answers
68 views

Expressing $ 12\sin( \omega t - 10) $ in cosine form

$$ 12\sin( \omega t - 10) $$ I understand how it's solved when using the graphical method, however I'm having trouble understanding something about the trigonometric identities method. The solution ...
2
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6answers
299 views

Range of a trigonometric function

Question: Prove that: $$0 \leq \frac{1 + \cos\theta}{2 + \sin\theta}\leq \frac{4}{3}$$ I have absolutely no idea how to proceed in this question. Please help me!
3
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2answers
40 views

Solving for an angle

I was never good in trigonometry. I have a rectangle with dimensions $L_1$ and $W_1$. I want to rotate it so that it fits inside another rectangle with dimensions $L_2$ and $W_2$. I need to find the ...
5
votes
5answers
526 views

Evaluate $\int_0^{{\pi}/{2}} \log(1+\cos x)\, dx$

Find the value of $\displaystyle \int_0^{{\pi}/{2}} \log(1+\cos x)\ dx$ I tried to put $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but I am unable to proceed further. I think the following integral can be ...
0
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1answer
32 views

Use $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial?

I am reading Barbeau's Polynomials and I found the following problem: Use the identity $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial. But I really have no idea on how ...
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2answers
55 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
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1answer
341 views

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is

In a triangle ABC,a:b:c is 4:5:6.The ratio of the radius of the circumcircle to that of incircle is
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2answers
38 views

When can I and when can I not use complex replacement?

If I want to calculate: $$(2 cos(t))^3$$ Can I not replace cos(t) with $Re(e^{it})$ and calculate $(2e^{it})^3$ to be $8e^{3it}$ and thus the real part of this becomes 8cos(3t)? But that answer is ...
1
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2answers
41 views

Rearranging equation $t = \frac{T}{2\pi} (\psi - \epsilon \sin \psi)$ in terms of $\psi$

I was playing around with the maths for orbits and trying to make a parametric equation that, well.. worked. I found a worksheet with parametrics with another variable ($\psi$), but I wanted to be ...
1
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1answer
59 views

Is $f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$ strictly positive?

I would like to have an advice for this exercise. Let $x\in[0,\pi]$ For which values of $x$ this function $$f(x)=\frac{\sin(x)}{\cos(2x)}+\sin(x)-\cos(x)$$ is strictly positive ? I tried to ...
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3answers
89 views

Implicit differentiation of trig functions

I'm struggling somewhat to understand how to use implicit differentiation to solve the following equation: $$\cos\cos(x^3y^2) - x \cot y = -2y$$ I figured that the calculation requires the chain ...
2
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2answers
181 views

show that $ \lim(\sin(\frac{1}{x}))$ as $x$ approaches zero does not exist.

I am stuck trying to understand how to prove that the limit of $\sin(\frac{1}{x})$ as $x$ approaches $0$ does not exist. a hint was given: a limit does not exist if there exists an $\varepsilon > 0$...
0
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0answers
20 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to [duplicate]

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to note s=a+b+c/2 area of triangle rs/2
3
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3answers
896 views

In a triangle ABC, (b + c) cos A + (c + a) cos B + (a + b) cos C is equal to

In a triangle $ABC$ $$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C=?$$
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2answers
81 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that $$\mathrm{area\,of\,triangle}=\sqrt{s(s-a)(s-b)(s-c)}$$...
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0answers
76 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
3
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5answers
105 views

If $\sin( 2 \theta) = \cos( 3)$ and $\theta \leq 90°$, find $\theta$

Find $\theta\leq90°$ if $$\sin( 2 \theta) = \cos( 3)$$ I know that $\sin 2\theta = 2\sin\theta\cos \theta$, or alternatively, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$. Can somebody help me?
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1answer
200 views

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle, then P^-1+P^-2+P^-3 is equal to

If P1, P2, P3 are altitudes of a triangle ABC from the vertices A, B, C and is the area of the triangle , then P^-1+P^-2+P^-3 is equal to
0
votes
2answers
285 views

Vectors, How to measure total force and direction.

I am currently looking for some math help that I am quite struggling with. The problem is: (Vectors) A fisherman use his pole and line to pull a fish out of the water. The line exerts a force on ...
1
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0answers
35 views

Finding the widest angle to shoot a soccer ball from the sideline using optimization!! [duplicate]

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. As ...
1
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2answers
59 views

Trying to solve a trig identity

Given $\sin 2x+\sin x=1$, find the value of $\cos 2x+\cos 4x$. I know $\cos 2x+\cos 4x \implies 1-2\sin^2x+1-2\sin^22x$, but didn't get the answer.
3
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1answer
640 views

Squeeze Theorem: Finding the limit of a trig function

I'm stuck on finding the limit of a complex fraction/trig function. Could someone please assist, or point out where I'm going wrong? Determine $$\lim\limits_{x \to 0} \frac{(x+1)\cos(\ln(x^2)...
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2answers
76 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over 1+x^2\right)}\...
0
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6answers
134 views

Why $\cos^2 x-\sin^2 x = \cos 2x\;?$

I was hoping someone could explain how $\cos^2 x-\sin^2 x = \cos 2x$ After using the product rule to differentiate $\sin x \cdot \cos x$ I get the answer $\cos^2 x - \sin ^2 x$ I've come across this ...
0
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1answer
38 views

Differentiate $(x-1)^2 \sin x$ where $x$ is in radians

How would I differentiate, simplify and then find $f'(\pi/2)$: $$ f(x)=(x-1)^2 \sin x $$ I'm not sure how to differentiate $\sin x$ to then use it later to find an answer, any help would be much ...
1
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3answers
118 views

Proving $\sin^2x+\cos^2x=1$ [duplicate]

I'm looking for a way to prove $\sin^2x+\cos^2x=1$ without using the Pythagorean Theorem or proving it. The only thing I've tried was to play with the Taylor series, but I didn't get far with that as ...
25
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2answers
428 views

Disproving an “almost true” trigonometric identity

The plausible looking "identity" $$\sin(\frac{\pi}{51})+\cos(\frac{\pi}{74})=\frac{3}{2\sqrt 2}$$ is not true, but it is close indeed: $$LHS=1.0606\color{blue}{598...}$$ $$RHS=1.0606\color{red}{601....
2
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1answer
102 views

How to solve $\tan x=x^2$ in radians?

How to solve $\tan x=x^2$ with $x \in [0, 2\pi]$? I try with trigonometry and many ways but the numerical solutions seems to be difficult.
2
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0answers
107 views

Evaluating $\int_0^x \lvert \cos t \rvert dt$

in my mathbook there is given a solution to $$\int_0^x \lvert \cos t \rvert \, dt $$ but without any hints or tips. $$\int_0^x \lvert \cos t \rvert \, dt = \sin\left(x - \pi \left\lfloor \frac x \...
0
votes
1answer
42 views

Which is greater and why?

This question is supposed to be conceptual and I am to determine which is greater without a calculator. So what is greater: cos (26 degrees) or cos (27 degrees). I am not sure how to explain why it ...
3
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1answer
101 views

Rational values of $\sin(\log(x))$

Apart from the trivial solution $\sin(\log(1))=0$, is $$\sin(\log(x))$$ ever rational if $x$ is rational?
1
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2answers
105 views

Use the range for $\theta$ to determine the indicated function value

Problem: $$\sin\theta = \frac {1}{2}; \frac {\pi}{2} \leq \theta \leq \pi, \text{find} \cos\theta$$ So I have that $\sin\theta = \frac {1}{2}$ and that the range for $\theta$ is $\frac {\pi}{2} \...
7
votes
2answers
491 views

Is $f(x)=10$ a periodic function?

I am not getting satisficatory explanation for this. Clearly $f(x+T) = f(x)$ for all values of $T$. If we assume it is periodic, does this mean period = $0$?
1
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2answers
338 views

Evaluating an inverse function by sketching a unit circle

Problem I'm working on: "Evaluate the inverse function by sketching a unit circle, locating the correct angle and evaluate the ordered pair on the circle." The function I got was $\cos^{-1}(0)$. ...
3
votes
2answers
249 views

Finding the limit of a trigonometric function

Could someone please clarify whether my calculation on the following limit problem is correct? Determine the following limit: $\lim_{x \to \frac{\pi}{2}} \frac{\sin^2x-1}{\sin x-1}$ $\lim_{x \to \...
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2answers
96 views

How solve this: if $\sin{(ax+b)}=\sin{29x}$ for all integers $x$, find the smallest possible value of $a$

Question: Assume that $a$ and $b$ are nonnegative real numbers such that $$\sin{(ax+b)}=\sin{29x}$$ for all integers $x$. Find the smallest possible value of $a$. This problem is from a ...
2
votes
1answer
67 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
3
votes
4answers
72 views

Show that $(7\cos(x)-\sin(x))^2=A\cos(2x)+B\sin(2x)+C$ for some integers $A,B,C$

How do you solve this question?:$$(7\cos(x)-\sin(x))^2=A\cos(2x)+B\sin(2x)+C$$ is for all $x$. Here $A$, $B$ and $C$ is constants. I need to know $A$, $B$ and $C$ to pass this. They are integers. I ...
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3answers
156 views

Solve the length AB (the dashed line)

Can someone show me how I can solve this? (Step by step example with solution appreciated a lot as I am currently practicing). EDIT: After a closer look, it looks as if this is an Isosceles ...
1
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2answers
97 views

If $\sin^2 \theta + 2\cos \theta – 2 = 0$, then find the value of $\cos^3 \theta + \sec^3 \theta$ [closed]

If $\sin^2 \theta + 2\cos \theta – 2 = 0$, then find the value of $\cos^3 \theta + \sec^3 \theta$.
1
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0answers
95 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) +(\frac{...
3
votes
2answers
106 views

If $ \dfrac{a^2-b^2}{a^2+b^2} = \dfrac{\sin(A-B)}{\sin(A+B)} $, then what type of triangle is $\triangle ABC $?

In $\triangle ABC$ $ \dfrac{a^2-b^2}{a^2+b^2} = \dfrac{\sin(A-B)}{\sin(A+B)} $ then what type of triangle is $\triangle ABC $ ? My try : By componendo and dividendo $\dfrac{a^2}{b^2} = \tan ...
1
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2answers
221 views

What is the first step to solving $\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x)$?

My calculus BC teacher has given us some trig "review". $$\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x).$$ How do I get rewrite the cos3x and sin3x? Do I just use sum and difference, because it ...
2
votes
6answers
780 views

How to find $\lim_{x\to 0}\frac{\tan 3x}{\tan 5x}$?

I am asked to find the following limit: $$ \lim_{x \to 0} \frac{\tan 3x}{\tan 5x}$$ My problem is in simplifying the function. I followed two different approaches to solve the problem. But both ...
4
votes
2answers
50 views

if $a$ is are real number that $a \neq 0$, and $\cos x = \sqrt{\frac{\cot x}{\cot x -a^2}}$, $x$ is on which trigonometric quadrants?

if $a$ is are real number that $a \neq 0$, and $\cos x = \sqrt{\frac{\cot x}{\cot x -a^2}}$, $x$ is on which trigonometric quadrants? Things I have done so far: this problem is mostly different ...
0
votes
1answer
490 views

Find the acute angle that has the same sin, cos, and tan values as…

$5\pi$ radians, $-450$ degrees, $\frac{11}{2}\pi$ $5\pi$ radians is exactly the same as $\pi$ radians, but $\pi$ is not an acute angle. How should I answer this question? Also, $-450$ degrees is ...
6
votes
5answers
369 views

How to find the polynomial such that …

Let $P(x)$ be the polynomial of degree 4 and $\sin\dfrac{\pi}{24}$, $\sin\dfrac{7\pi}{24}$, $\sin\dfrac{13\pi}{24}$, $\sin\dfrac{19\pi}{24}$ are roots of $P(x)$ . How to find $P(x)$? Thank you very ...
2
votes
4answers
92 views

What is the maximum value of $ \sin x \sin {2x}$

What is the maximum value of $$ \sin x \sin {2x}$$ I have done my work here $$f (x)=\sin x \sin 2x =\frac{\cos x - \cos3x}2 $$ $$f'(x)= \frac{- \sin x+3 \sin 3x}2 =4\sin x (2-3\sin^2 x)=0$$ $$x=0,\pi;...
0
votes
2answers
210 views

What is the y-cooridinate for the point on the curve with x-cooridante 20?

What is the y-coordinate for the point on the curve with x-coordinate 20? $F. 160$ $G. 162$ $H. 164$ $J. 166$ $K. 168$ The explanation says "The correct answer is G. If the x-coordinate is 20, ...
7
votes
3answers
102 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.