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

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3answers
28 views

Value of Sine Function from data given [on hold]

If $0 \le \alpha , \beta \ge 90\ $ and $ \tan(\alpha - \beta) = 2$ and $\tan (\alpha + \beta) = 3 $, Then what is the value of $\sin 2\beta$ ?
1
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4answers
64 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
0
votes
1answer
48 views

Maximize the trigonometric expression

Find the maximum value of $$4\sin^2 x+3\cos^2 x+\sin(x/2)+\cos(x/2)$$ Please give some hints. I tried writing the angles in half-angles but it didn't help. Thanks.
-3
votes
2answers
44 views

How to calculate an elementary integral

How do you calculate $$\int\dfrac{2 du}{(u^2+1)^2}$$ It does not seem too difficult but I do not know which method to use.
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1answer
28 views

Help with primitive function

I need help evaluating the indefinite integral $$\int\frac{\cos(5x) + \cos(4x)}{1-2\cos(3x)}dx.$$
4
votes
1answer
62 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
0
votes
0answers
12 views

Find Intersection of Two Circle given Lat/Lon and radius

I am attempting to calculate the intersection of two circle on the Earth with a given latitude, longitude and radius. I started with this post. While I am using this in the context of programming, it ...
2
votes
7answers
42 views

Some help on trigonometric equation

So I have $\sin^3x = \frac 34 \sin x$. Can you expand so the answer is either $\sin x(\sin^2x +\frac 34)$ which leads to the answer $\frac 12 + 2n\pi$ or that $\sin^3x = \frac 14(3\sin x-\sin^3x) - ...
0
votes
3answers
38 views

Find the number of integral values of $k$ if $ \sin4x - \cos4x + 3 \sin2x= k$

Please help me with this question How to find the number of integral values of $k$ that satisfy the given equation: $$ \sin4x - \cos4x + 3 \sin2x= k$$ My attempt: On solving the above equation, the ...
1
vote
1answer
24 views

Evaluating sin using sum/diff identities

EDIT: IM DONE WITH THIS PROBLEM, THANKS FOR THE HELP Evaluate the expression under the given conditions. My work (got lost and don't know what to do from here): EDIT: Sin theta should be -3/5 ...
0
votes
1answer
29 views

Sum/Difference Identity Formula Question

Wouldn't this be, per the sum/difference identity formula, $\cos (\frac{13\pi}{5}-\frac{\pi} {5})$ which is $\cos (\frac{12\pi}{5})$?
0
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0answers
48 views

Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undifend period "$d$". So for every $d\ge2$. $$\zeta(-s)= \frac{1}{(d^{s+1}-1)}\sum_{m=1}^{\infty} \frac{1}{2^{m+1}}\sum^{m}_{j=1} ...
1
vote
1answer
37 views

'Chasing sides' in a geometry problem

Consider the circle $W=x^2+y^2=81$. Let $AB$ be a diameter of circle $W$. $AB$ is extended through $A$ to $C$. Point $T$ lies on $W$ so that line $CT$ is tangent to $W$. Point $P$ is the foot of the ...
1
vote
2answers
54 views

Trying to solve a pair of trigonometric simultaneous equations

I have a machine that has two shafts which are the inputs and their position is set by 2 servo motors. Depending on the angle of these two shafts (shaft 1 has an angle designated $Ta$ degrees, shaft 2 ...
-1
votes
2answers
50 views

How to find sin and cos of 0, pi/2, pi without calculator [on hold]

In my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate.
2
votes
0answers
55 views

proving a inverse trigonometric expression

Show that $2 \tan^{-1}\frac{\sqrt{x^2+a^2}-x+b}{\sqrt{a^2-b^2}}+\tan^{-1}\frac{x\sqrt{a^2-b^2}}{b\sqrt{x^2+a^2}+a^2}+\tan^{-1}\frac{\sqrt{a^2-b^2}}{b}=n\pi$ I tried algebric simplification,but was ...
0
votes
0answers
26 views

Rotating one coordinate system about another

I have two coordinate systems: A and B. I also have a point p, whose position relative to ...
1
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2answers
56 views

Prove that type question of Trigonometric Identities

If $3\sin A + 5\cos A = 5$, then prove that: $$5\sin A + 3\cos A = ±3.$$
1
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2answers
68 views

Is $ \cos² y = 0 $ a solution?

I'm studying math for school. We're solving separable differential equations. One of the exercises is: $$ \frac{\Bbb d y}{\Bbb d x} = \frac{ (\cos y)^2 \tan y }{1+x²}$$ If you separate the ...
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votes
0answers
28 views

2 non right-angles triangles. [on hold]

I have two non right-angled triangles stacked together. The small one is inclined at an angle of $30^{\circ}$ with the horizontal and a vertical height of 30m. The other triangle has two adjacent ...
1
vote
1answer
34 views

Does there exist a $z\in \Bbb R$ such that $\sin z=t \in \Bbb T$?

Does there exist a $z\in \Bbb R$ such that $\sin z=t \in \Bbb T$: the set of transcendental numbers? I've had this doubt and I didn't know how to tackle it... Edit: Changed my domain to reals only, ...
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vote
2answers
54 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
0
votes
2answers
55 views

Max value of trignometric function $\sin \left(x+\frac\pi6\right)+\cos \left(x+\frac\pi6\right)$

Question: The maximum value of $\sin \left(x+\dfrac{\pi}{6}\right)+\cos \left(x+\dfrac{\pi}{6}\right)$ is at what value of $x$. I solved the problem by setting the slope of the function to zero and ...
3
votes
1answer
46 views

Let $S$ be a set of $n$ points in the plane with min spacing of 1. Prove $S$ has a subset of $\ge n/7$ points with min spacing of $\sqrt{3}$.

I believe I have proven the case $n=8,|T|=2$, but welcome feedback. I need help proving the case for general $|T|>2$. From the 2003 Canada National Olympiad: Let $S$ be a set of $n$ points in ...
2
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1answer
47 views

Evaluating the indefinite integral $\int\sqrt{\cos2x}\sin^32x\,dx$

I have tried to integrate the following indefinite integral but I'm not sure if I get the right answer. Please tell me if I'm wrong and if so, please indicate what went wrong. $$ ...
2
votes
4answers
44 views

Find all values that solve the equation

For which values a, the equation $$ a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 $$ has a solution? My idea: I think it's possible to factorize equation or reduce equation to the ...
1
vote
1answer
42 views

Prove that a trigonometric equation has six distinct roots

Show that,in general,the equation $A \sin^3x+B\cos^3x+c=0 $has six distinct roots,no two of which differ by $2\pi$,and that the tangent of their semi-sum is $-\frac{A}{B}$. My attempt: I tried to ...
1
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2answers
58 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
3
votes
2answers
38 views

Prove the relation for cos inverse

Prove the relation $\cos^{-1}x_0=\dfrac{\sqrt {1-x^2_0}}{x_1\cdot x_2\cdot x_3\cdots \text{ ad inf.}}$ where the successive quantities $x_r$ are connected by the relation ...
1
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1answer
40 views

Minimum value of trigonometric function

The minimum value of the expression $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\mathrm{cosec} x\right|$ can be expressed as $(\sqrt a-\sqrt b)$ where a and b are natural number then find the value of ...
0
votes
2answers
33 views

Unique solution to sin(2a) and cos(2a)

I'm a bit confused as to how to solve for $2\alpha^*$ using the equation 6.36 in the excerpt below. I know how to solve for it individually (ie acos and asin) but how do I solve them together to get ...
1
vote
2answers
44 views

Finding a triangle ABC if $2\prod (\cos \angle A+1)=\sum \cos(\angle A-\angle B)+\sum \cos \angle A+2$

Find $\triangle ABC$ if $\angle B=2\angle C$ and $$2(\cos\angle A+1)(\cos\angle B+1)(\cos\angle C+1)=\cos(\angle A-\angle B)+\cos(\angle B-\angle C)+\cos(\angle C-\angle A)+\cos\angle A+\cos\angle ...
3
votes
1answer
78 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
0
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1answer
11 views

Find point on circle's tangent based on point on circle, radius and angle

The circle is centered at (0,0)"P" with a radius of 5. I have a point on the circle at (4,-3)"A". How would I find the points "B1" and "B2" on the tangent through point "A" given an arbitrary angle ...
1
vote
3answers
64 views

how can i prove this trigonometry equation

I need help on proving the following: $$\frac{\cos {7x} - \cos {x} + \sin {3x}}{ \sin {7x} + \sin {x} - \cos {3x} }= -\tan {3x}$$ So far I've only gotten to this step: $$\frac{-2 \sin {4x} \sin {3x} ...
1
vote
6answers
170 views

Sum of cosines of complementary/suplementary angles

Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero? Could you prove it by some identity?
9
votes
7answers
138 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
4
votes
7answers
298 views

Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
0
votes
2answers
22 views

ind $\tan \alpha$ in the square

let say the square has sides of 2 units, $DM = DN = AN = AP = 1$, $NP = \sqrt 2$, $NQ = QP = \frac{\sqrt 2}{2}$, and $AR \ne AP$ (?) we have know that $\tan \alpha = \frac 2{RP}$, but what's the ...
0
votes
3answers
17 views

Determine if 2 points are horizontal without trigonometry

Let's say that I have 2 points: (c1X, c1Y) and (c2X, c2Y). I would like to consider these 2 points horizontal as long as their angle is below 45 degrees. I could accomplish this with trigonometry. ...
2
votes
4answers
73 views

Find $x$ in the triangle

the triangle without point F is drawn on scale, while I made the point F is explained below So, I have used $\sin, \cos, \tan$ to calculate it Let $\angle ACB = \theta$, $\angle DFC = \angle ...
0
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1answer
18 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
0
votes
1answer
44 views

A triangle ABC with the internal bisector of $\angle A$, the median drawn from B and the altitude drawn from C meet at the same point.

A triangle $ABC$ with the internal bisector of $\angle A$, the median drawn from $B$ and the altitude drawn from $C$ meet at the same point. Prove that $$\tan A = \dfrac{\sin C}{\cos B}$$ I try to ...
1
vote
1answer
40 views

Geometric problem based on angle bisectors

I am not asking a question,i just want to conform,is my method of solving problem correct? Given a triangle ABC.It is known that AB=4,AC=2,and BC=3.The bisector of angle A intersects the side BC at ...
0
votes
2answers
43 views

Trig identity involving sum of cosines

$$\begin{align} y &= A\cos\left(\omega t - kx +\phi_1\right)+A\cos\left(\omega t + kx + \phi_2\right)\\[6pt] &= 2A\cos\left(\omega t+\frac{\phi_1+\phi_2}{2}\right)\cdot ...
0
votes
0answers
58 views

Find the value of $h$ from a Kepler-type equation

$$V = \frac{0.5r^{2}\cdot \cos^{-1}(\frac{r-h}{r})\cdot 2-\sin(\cos^{-1}(\frac{r-h}{r})\cdot 2)}{10^{6}}$$ This is the equation to find the volume of liquid in a tank in the shape of a capsule. Where ...
2
votes
3answers
21 views

Eliminate the parameter of a

Eliminate the parameter to find a description of the following circles or circular arcs in terms of $x$ and $y$. Give the center and radius, and indicate the positive orientation. ...
0
votes
1answer
21 views

Question based on incenter and excenter

In a $\bigtriangleup ABC $,$sin\frac{A}{2}+sin\frac{B}{2}+sin\frac{C}{2}=\frac{6}{5}$ and $II_1+II_2+II_3=9$ where I is incenter and $I_1,I_2,I_3$ are the excenters of $\bigtriangleup ABC $.Then find ...
1
vote
3answers
87 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
1
vote
2answers
55 views

Solve $2\sin^3x + \sin3x +3\sin^2x \cos x + \cos^3x=0$

$2\sin^3x + \sin3x +3\sin^2x\cos x + \cos^3x=0$ My try: $$2\sin^3x +3\sin x - 4\sin^3x +\cos x(3\sin^2x+\cos^2x)=0 $$ $$ \cos x(2\sin^2x+1) - 2\sin^3x+3\sin x=0.$$ And then i have no idea.