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
253 views

Can't solve this trignometric equation, why am I wrong?

There is this trig equation: $$ 5\tan x - 2\tan 2x = 0 \text{ for 0 < 0 < 360 } $$ So far I've gotten $$\tan x = \text{0, 180}$$ and all I have to solve now is $$\tan ^2x = 0.2$$ which gives ...
1
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2answers
54 views

Trigonometric Identities Need Help

I'm struggling with this. Can someone help me? I need to make $$-\sin(x)$$ into $$(\cot(x)-\csc(x))(\cos(x)+1)$$ Does anyone know the solution? Thank you.
3
votes
2answers
75 views

Find the Value of Trigonometric Expression

Given $$\begin{align} \frac{\cos \alpha}{\cos \beta}+\frac{\sin \alpha}{\sin \beta}=-1 \end{align} \tag{1}$$ Find the value of $$\begin{align} \frac{\cos^3 \beta}{\cos \alpha}+\frac{\sin ...
2
votes
1answer
27 views

Establishing an identity involving cotangent and cosecant

$$\frac{\csc(x)-1}{\cot(x)}=\frac{\cot(x)}{\csc(x)+1}$$ Once again, "Professor Google" provides an example that's different enough that I can't solve "my" problem. I'm beginning to think that Google ...
1
vote
2answers
64 views

Establishing the identity

$$\csc(x)- \cot(x)= \frac{\sin(x)}{ 1+ \cos(x)}$$ I'm completely stumped. There are a few examples with the signs reversed but this is just different enough that none of the examples work. Is this a ...
0
votes
3answers
139 views

To prove $\cos(A+B) = \cos A \cos B - \sin A \sin B$ [closed]

How to prove the formula $\cos(A+B) = \cos A \cos B - \sin A \sin B $ by using cross product of two vectors?
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3answers
548 views

In a triangle ABC, prove that cot(A/2)+cot(B/2)+cot(C/2) =cot(A/2)cot(B/2)cot(C/2)

In a triangle ABC, prove that $\cot \left ( \frac{A}{2} \right )+\cot \left ( \frac{B}{2} \right )+\cot \left ( \frac{C}{2} \right )=\cot \left ( \frac{A}{2} \right )\times \cot \left ( \frac{B}{2} ...
0
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2answers
132 views

Prove that $\small\sin x\sin y\sin(x-y) + \sin y \sin z \sin(y-z) + \sin z \sin x \sin(z-x) + \sin(x-y) \sin(y-z) \sin(z-x) = 0$.

Prove that $$\sin(x) \sin(y)\sin(x-y) + \sin(y) \sin(z) \sin(y-z) + \sin(z) \sin(x) \sin(z-x) + \sin(x-y) \sin(y-z) \sin(z-x) = 0 \; .$$ I tried all identities I know but I have no idea how to ...
2
votes
2answers
249 views

Prove that $\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18$ [closed]

Prove that $$\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18.$$ Without using a calculator. I tried all identities I know but I have no idea how to proceed. I always get stuck on finding ...
0
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0answers
70 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
-1
votes
1answer
215 views

Sine & Cosine Word Problem [closed]

Problem: While a student was playing with their calculator they found that sometimes the answers produced from taking the sine and cosine of different angles were the same answer. The information ...
1
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2answers
79 views

Finding the zeros of trionometric polynomails.

I have a question about something I've struggled with for a while: Finding the zeros of trigonmetric polynomials. Let me show you a problem I am solving and you guys can tell me if I got the right ...
1
vote
1answer
132 views

Determining intersecting points between square and circle

I unfortunately have spent too much time trying to solve this question, and have turned to you for help. The corner of my square has intersected some circle, and I need to move it out. I only know one ...
1
vote
1answer
184 views

Show that: $ (\csc\theta - \sin\theta)(\sec\theta - \cos\theta) \equiv \frac{1}{\tan\theta + \cot\theta}$

I am having hard time solving this question, I start with the L.H.S and my answer always boils down to $\sin\theta\cos\theta$ And I don't know what to do after that.
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votes
2answers
132 views

Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
1
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5answers
121 views

Prove $\frac{\sin A}{\sec A+\tan A-1}+ \frac{\cos A}{\csc A+\cot A-1}=1$

$$\frac{\sin A}{\sec A+\tan A-1}+ \frac{\cos A}{\csc A+\cot A-1}=1$$ Prove that L.H.S.$=$R.H.S. This type of questions always creates problem when in right hand side some trigonometry function is ...
1
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4answers
114 views

inverse trigonometric equation $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$

I have problem with showing that $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$ I think there have to be used formula: $\displaystyle ...
1
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2answers
132 views

Trigonometric Arithmetic Progression

If $a$, $b$, $c$ are in arithmetic progression, prove that $$\cos A \cot\frac{A}{2} \qquad \cos B \cot \frac{B}{2} \qquad \cos C \cot\frac{C}{2}$$ are in arithmetic progression, too. Here, $a$, $b$, ...
1
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1answer
120 views

Finite integral with goniometric functions, $\int_0^{\infty} \frac{8\sin^4(\pi f t)\tan^2(\pi f/2)}{(\pi^4 \tau^2 f^3) }df$

I have difficulties trying to find an algebraic solutions of the following integral: The $\tau$ in this formula is an integer, which is a very important fact because only then this integral is ...
2
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4answers
133 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
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2answers
66 views

Trigonometry question: $\sin^2(A) + \sin^2(B) - \sin^2(C) = 2\sin(A)\sin(B)\cos(C).$

Given $A + B + C = 180$, prove that $$\sin^2(A) + \sin^2(B) - \sin^2(C) = 2\sin(A)\sin(B)\cos(C).$$ I tried all identities I know but I have no idea how to proceed.
3
votes
9answers
204 views

Find $\tan x $ if $\sin x+\cos x=\frac12$

It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $. I tried all identities I know but I have no idea how to proceed. Any help would be appreciated.
3
votes
4answers
85 views

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$ $$\tan x + \sec x = 2\cos x$$ I tried changing it all to sin and cos $$\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2\cos x$$ then I ...
0
votes
1answer
83 views

Solve for x: sin2 x − cos2 x = sin x, −π ≤ x ≤π

I have to solve for x using the domain of −π ≤ x ≤π sin^2 x − cos^2 x = sin x I tried changing cos^2 x to 1 - sin^2 x so it would be sin^2 x - 1 + sin^2 x = sin x making it, 2sin^2 x - 1 = sin x ...
4
votes
4answers
8k views

How to memorize the trigonometric identities?

I am stuck trying to memorize the trig identities, and try as I may, I just can't get them to stick (especially the sum-product and product-sum formulas). I am concerned I won't be able to memorize ...
3
votes
3answers
63 views

Trigonometry Question: find Value of…

Find value of $3 + \cos2x + \cos4x + \cos6x - 4\cos x\cos2x\cos3x$. I tried with $\cos A + \cos B$ identity but it was not simplifying.... Help..
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3answers
408 views

please help me. ive been trying to solve this for hours [closed]

At 3:00 PM, a boat is 12.5 miles due west of a radar station and traveling at 11 mph in a direction that is 57.3 degrees south of an east-west line. At what time will the boat be closest to the radar ...
1
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5answers
278 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
0
votes
1answer
39 views

Acute angle and trigonometric functions [closed]

Given that $\theta$ is an acute angle and $\cos\theta = \dfrac{7}{25}$. Find: $\tan\theta$, $\sin\theta$, $\sec\theta$.
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0answers
39 views

The roots of transcendent equation $\tan(x)=x$ [duplicate]

Can we find the roots of equation $\tan(x)=x$. I once found a formula which gives its roots approximately. Any link will be wlecomed.
3
votes
2answers
145 views

Calculation for the chance of finding something a given distance from a starting point by walking straight in a random direction?

The premise is basically a 2D plane with a single point, the starting point. Now a landmark sought by a hiker is a certain distance from that point. If the hiker can only see 1 mile in any ...
2
votes
1answer
97 views

Trigonometric Substitution and the Triangle Inequality

I was reading the solution to this problem: If $x, y, z$ are real numbers and $x+y+z=xyz$ then $x(1 − y^2 )(1 − z^2 ) + y(1 − z^2 )(1 − x^2 ) + z(1 − x^2 )(1 − y^2 ) = 4xyz$ The solution is to ...
6
votes
4answers
757 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 $\frac1{\sqrt2}$ ...
1
vote
1answer
54 views

Proving the following trigonometric proportion

$$\frac{a\sin(B-C)}{b^2-c^2}=\frac{b\sin(C-A)}{c^2-a^2}=\frac{c\sin(A-B)}{a^2-b^2}$$ A, B, C are angles of a triangle and a, b, c are the sides of a triangle. I tried using various things such as ...
1
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2answers
81 views

Trigonometric relation between 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.
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1answer
124 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?
3
votes
3answers
95 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} - ...
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3answers
262 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 ...
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3answers
53 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 ...
0
votes
3answers
452 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 ...
11
votes
2answers
336 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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1answer
42 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
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4answers
88 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 ...
0
votes
3answers
163 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?
4
votes
2answers
93 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 ...
1
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3answers
59 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 ...
12
votes
3answers
334 views

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

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
0answers
76 views

Solving systems of equations with trigonometric terms

I am trying to solve (or rather find the least squares solution for) a system of equations with trigonometric terms in them. The system consists of pairs of equations of the form $a_1 \cos\theta - ...
2
votes
2answers
68 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 ...
2
votes
2answers
79 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 ...