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

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1answer
28 views

Proving the following proportion

$$\frac{a\sin(B-C)}{b^2-c^2}=\frac{b\sin(C-A)}{c^2-a^2}=\frac{c\sin(A-B)}{a^2-b^2}$$ I tried using various things such as sine rule and then replacing the various rations in terms of sides if ...
4
votes
4answers
127 views

Double Angle Equations

$\cos2x=\frac1{\sqrt2}$ is the original problem, and I have to solve for $x$. However, I'm not sure what to do after I substitute the double angle formulas for $\cos2x$. I know that $\frac1{\sqrt2}$ ...
0
votes
1answer
19 views

Sides and angles of a triangle

$$a \cdot \sin (B-C) +b \cdot \sin(C-A) +c \cdot \sin(A-B) =0$$ where $a, b, c$ are the sides of a triangle and $A, B, C$ are the angles of a triangle No idea how to solve this problem
0
votes
0answers
17 views

Expand trigonometric expression

I am supposed to expand this expression $${\frac {\sin \left( x \right) b \left( 4\,b\cos \left( x \right) + \sqrt {16\,{b}^{2}+1}+5 \right) }{4\,b\cos \left( x \right) +\sqrt {16 \,{b}^{2}+1}+1}} $$ ...
0
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1answer
13 views

Find points of triangle, one point, all sides and all angles known

Imagine the setup above; how can I calculate the points P1 and P2 if all angles, all sides A,B,C and point P3 are known?
1
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1answer
34 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
0
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2answers
44 views

Expressing the sine function in terms of exponential

Prove $e^{iz} - e^{-iz} = \sin z$. I used $$\begin{align*} \sin z & = z - z^3/3! + z^5/5! - z^7/7! + \dots & (i) \\ e^{iz} & = 1 - z^2/ 2! - iz^3/3! + \dots & (ii) \\ e^{-iz} ...
-4
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3answers
68 views

Finding the sine of an angle formed within a square by two midpoints and a vertez [closed]

Let $abcs$ be a square, and suppose that $an=nd=dm=mc$ as on the diagram. Find $\sin x $
3
votes
3answers
57 views

What is the value of $ \int_{x}^{1} \arcsin \left( \frac{2t}{t^2+1} \right) \text{d}t $?

Is this result true? Wolfram doesn't seem to be able to evaluate the definite integral in the allowed time. $$ \int_{x}^{1} \arcsin \left( \dfrac{2t}{t^2+1} \right) \text{d}t = \dfrac{\pi}{2} - ...
0
votes
3answers
108 views

Simplify tan$\theta$ cos$\theta$

How do I simplify tan$\theta$ cos$\theta$ ? Why is this so hard to do? What pieces of information should I know before doing these? Can someone just tell me were am I going wrong? I have 5 days ...
6
votes
2answers
176 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
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vote
3answers
52 views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)1\})$ and determining its dimension.

We consider a space F(R,R) of functions of R in R. Let A = ({1, \sin(x), $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the identity ...
2
votes
1answer
42 views

What trig identities must one know to derive the others?

My TA told me in problem section one day that every trig identity could be derived from just 2: the Pythagorean identity and the double-angle identity (or he might have said the half-angle identity). ...
0
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3answers
37 views

Check my answer for this trigonometric identity

Simplify $\sec x \cot x$ $\sec x = \frac{1}{\cos \theta}$ $\cot x = \frac{1}{\tan \theta}$ therefore $\frac{1}{\cos \theta} · \frac{1}{\tan \theta} = 1(\tan\theta) + 1(\cos\theta) = \frac{\sin ...
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3answers
50 views

Simplify $\tan(360 - \theta)$

I am aware that $\tan(\alpha-\beta)=\dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)}$ So for my question: $\tan(360 - \theta)$ Do I choose random value for $\theta$ and plug it into the ...
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3answers
55 views

Simplify $\sin (90 - \theta)$

Title. I have no idea what to do. Is their an identity I have to remember? What am I supposed to do to the equation? Do I have to solve for something first, what does it mean by simplify?
2
votes
2answers
47 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
0
votes
2answers
36 views

Trigonometric Identities help

How do you solve this? I can't figure out what I should do. $$\sin ^4\left(A\right)+\cos ^2\left(A\right)=\cos ^4\left(A\right)+\sin ^2\left(A\right)$$ Also, why is this equal zero? Can someone ...
0
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0answers
31 views

An application of squeeze theorem to the limit of $(\sin x)/x$ [duplicate]

How can I solve the following problem using elementary trigonometry? Use the Squeeze Theorem to conclude that $$\lim_{x\to 0^+} \frac{\sin(x)}{x} =1.$$ Use symmetries of $y = \frac{\sin(x)}{x}$ to ...
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2answers
28 views

The bird pointer problem: finding the angle of rotation

Suppose we have a bird pointer. He is a guy that likes to point at birds in the sky: His legs cannot move, however he can rotate around his torso. Also, his body and his arm always make a right ...
0
votes
1answer
29 views

Integer Solutions to Cosine's Dot Product Formula

Say one wanted to test their students on the dot product formula without a calculator. One would (being a nice teacher and all) natural like to pick numbers in the plane that are "nice" and satisfy a ...
0
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0answers
28 views

Does every triangle have a slope? [on hold]

I have two numbers or two digits to add. One is 0.84 and the other 0.16 both equal to one.If I divide numbers $\frac{1}{0.84}$ and $\frac{1}{0.16}$, subtract one to both results, I would get ...
12
votes
3answers
149 views

How to evaluate $\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$

Evaluate $$\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$$ I have found the problem on this page. I have no idea how to do it. Thank you very much.
2
votes
2answers
74 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
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1answer
25 views

Trigonometry - SSS formula to calculate angle of Triangle

I am trying to calculate angle using Law of cosines, SSS formula of Triangle. That is given three sides of triangle, calculate angle between sides. When there is large difference between all sides ...
2
votes
2answers
93 views

How am I supposed to work this out, or do I have to memorize?

When simplifying a trigonometric expression, say, $\sin^2 \theta$ / $\cos^2 \theta$ - I remember that sin over cos is equal to tan. However, what other identities, such as the one mentioned above, ...
4
votes
1answer
22 views

Family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$

Problem : If $\sin(\alpha + \beta)\sin(\alpha -\beta) =\sin\gamma (2\sin\beta +\sin\gamma), 0 < \alpha , \beta ,\gamma <\pi$ then the family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$ ...
0
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2answers
59 views

the period of a trigonometric function

I'm trying to solve a differential equation which is : $$y'(t)-4y(t) = \cos(3t)$$ Resolution of the equation without the second membre $y'(t)-4y(t)=0$ has as solution $ y_s(t)=ke^{4t} $ with ...
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2answers
30 views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
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0answers
27 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
3
votes
1answer
33 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
0
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0answers
13 views

Steps involved in simplifying trigonometric identities.

I am trying to master the simplification of trigonometric identities. When I look at a problem, asking me to simplify a trigonometric expression, I am not really sure what to do - but I do sort of ...
-3
votes
2answers
40 views

find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for sin(2θ+θ) [on hold]

Find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for $\sin(2 \theta+\theta$)
6
votes
7answers
141 views

Value of $\cos^2\alpha-\sin^2\alpha$

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 8: b) If $\tan\alpha=r$, write an expression in terms of $r$ that represents the value of $\cos^2\alpha-\sin^2\alpha$. ...
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2answers
39 views

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula.

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) $\cos^2 \left( \cfrac{θ}{2} \right)− \sin^2 \left( \cfrac{θ}{2} \right)$ (b) $2 \sin \left( ...
0
votes
1answer
20 views

Conversion of angle from 360 degree to-90 degree

Here i am trying to convert angle into +90 degree AND -90 DEGREE FORMAT.For desired elevation angle i got answer properly.How to converert angle -90deg to zero, zero to +90 degree format. you can ...
2
votes
4answers
2k views

Convert trig of angle in degrees to decimal value without a calculator

Been forever since I did trig, I know how to use my calculator to do it, but I can't remember if there is a way to evaluate $\cos(x)$ without a calculator. For example: $\cos(30)$ evaluates to ...
20
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5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
1
vote
2answers
54 views

Trignometric functions, Maximum value

$f(x)=\sin^{16}(x) +\cos^{18}(x)$, find the maximum value of the given function $f(x)$. I tried differentiating the given function however couldn't get the value of $x$ for which I could substitute ...
1
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0answers
31 views

Websites for math tests/quizzes

Next semester I'm taking calculus at college and I was looking for websites that have quizzes/test for things like trigonometry, trig formulas, pre-calculus, calculus readiness, etc. so I can get ...
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3answers
44 views

Solving Trigonometric Problems Like These

I am not sure how to approach this problem at all. I have no idea where to start or what it wants from me. Find the exact value of $\sin \theta$ and $\cos \theta$ given that $\cos ...
3
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2answers
53 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
2
votes
3answers
137 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
2
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1answer
51 views

Product-to-sum trigonometry identity

I'm really not sure about this Product-to-sum identity on wiki. See: I cannot find this anywhere on the web - does anybody know a reference? Certainly the one wiki gives does not cover it. I'm ...
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2answers
37 views

How to solve $3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$

I have got a bunch of trig equations to solve for tomorrow, and got stuck on this one. Solve for $\theta$: $$3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$$ I tried using ...
2
votes
5answers
69 views

Period of $\sin(x) + \cos(x)$

The period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$. And the period of $\sin(x)+\cos(x)$ is also $2\pi$. Why it is so?
3
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2answers
53 views

How to find the maximum value of $12\sin x -9\sin^2x$

How to find the maximum value of $12\sin x -9\sin^2x$ ; My approach : This can be written as $-[(3\sin x -2)^2-4]$. It means that the function will be maximum when $(3\sin x-2)^2 <4$ due to ...
0
votes
1answer
32 views

Triangulation in delayed loudspeaker setup

I could use some help with the following situation. Two physically displaced speakers need to arrive on time, in order to achieve summation, for a given position (green dot) within a listening plane ...
3
votes
1answer
183 views

Solving this trigonometric equation

$$\sqrt{3} \cos x - 3 \sin x = 4 \sin 2x \;\cos 3x$$ I tried many things: opening $\sin 2x$, $\cos 3x$, simplifying LHS: $\cos(60^\circ+x)$. Nothing seems to work. Any hint?
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2answers
41 views

Converting $5\sin 60^\circ$ to $5\sqrt{3} / 2$.

$$\eqalign{\dfrac x5&=\sin60^\circ \\ x&=5\sin60^\circ\\&=\dfrac{5\sqrt{3}}{2}}$$ Can someone tell me how the last part was derived? How do I get from $5\sin 60^\circ$ to $5\sqrt{3} / 2$? ...