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
24 views

Eliminate $A$ from these two equations.

$$x = \cot A + \tan A$$ $$y = \cos A + \sec A$$ Eliminate $A$ from these two equations. We tried doing $y^2 - x^2$ but it did not eliminate $A$.
0
votes
3answers
58 views

Trigonometry - Find the exact value of $\sin15^\circ$

I am having problems understanding how to solve $\frac{1}{4}(\sqrt6 - \sqrt2)$, find the exact value of $\sin15^\circ$. I have the answer, but I need help understanding the methods to achieve the ...
0
votes
2answers
27 views

Derivation of sine and cosine case

I am struggling to see this. I know that we can factor out $ a$, but I don't see how we can end up with the right hand side. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
1
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0answers
9 views

Trig equation that fits the plot points (octagonal pyramid)

I'm looking for an equation that satisfies these conditions: Input 90 degrees, result is 90 degrees Input 45 degrees, result is 60 degrees Input 0 degrees, result is 45 degrees For an input value ...
1
vote
1answer
18 views

How could I calculate the local size of an object given its distance and actual size?

Lets say I take a picture of a sign. I know that sign is 20ft (width), 10ft height. I'm standing 40 feet away. If I were to take a picture, how could I calculate how wide and how high the sign is in ...
2
votes
4answers
52 views

Show that there is an angle $\theta$ such that $a=\cos\theta$ and $b=\sin\theta$

My problem is from Israel Gelfand's Trigonometry textbook. Page 50. Exercise 3: Suppose that $\alpha$ is some angle. If $a=4\cos^3\alpha-3\cos\alpha$ and $b=3\sin\alpha-4\sin^3\alpha$, show that ...
1
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4answers
39 views

Trignometry-Prove that $(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$

Prove that $$(\csc\theta - \sec\theta )(\cot \theta -\tan\theta )=(\csc\theta +\sec\theta )(\sec\theta ·\csc\theta -2)$$ I tried solving the LHS and RHS seperately but they were not coming out to be ...
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1answer
46 views

Trignometry -For all $x$ in $[0,90]$ show that $\cos(sin(x)) >\sin(cos(x))$

For all $x$ in $[0,90]$ show that $\cos(sin(x))$>$ sin(cos(x))$ I understood the solution given in my book which said  $$\cos(x)+\sin(x)\leq\sqrt{2}<90$$ $$\cos(x)<90-\sin(x)$$ But if ...
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4answers
38 views

How to find an angle (in degrees) in a right triangle, given its sides?

I need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this ...
1
vote
3answers
28 views

Basic question about angles

Why is the answer a)? Why can't it be d)? Why are the choices listed in this format, i.e., $(x \pm \theta^{\circ})$, and why is angle C $(x+30^{\circ})$ and not just $30^{\circ}$? Thanks.
6
votes
4answers
534 views

A strange trigonometric equation

Today,in our class, we received a trigonometric equation $$\sin^{10}{x}+\cos^{10}{x}=\frac{29}{16}\cos^4{2x}$$ and the question was to find the general solution of this equation. My approach was, at ...
2
votes
3answers
168 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
5
votes
5answers
72 views

Simplify $2 \sin(x) \cos(7x) + \sin(6x)$

I was doing a problem and in my chain of computations I arrived at a seemingly complicated function $$2 \sin(x) \cos(7x) + \sin(6x)$$ However, I typed it into Wolfram and was surprised to find $$2 ...
3
votes
3answers
49 views

The average value of the function $y=\tan(2x)$ over the interval $[0,\frac{\pi}{8}]$

I was given the following question in a technology free exam. How would one go about solving this without the use of a calculator? NB. I am currently in my last year of high school so please don't ...
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votes
1answer
37 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
8
votes
2answers
133 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
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0answers
27 views

Definite Integration.Trigonometric function [on hold]

How to integrate $$3\sqrt { \cos ^{ 2 }{ \left( t \right) \sin ^{ 2 }{ \left( t \right) +\sin ^{ 4 }{ \left( t \right) \cos ^{ 2 }{ \left( t \right) } } } } } $$ for $t\epsilon \left[ 0,2\pi ...
0
votes
3answers
40 views

Help needed verifying a trigonometric identity

I have the following identity: $$ \frac{\tan (t + h) - \tan(t)}{h} = \left( \frac{\tan (h)}{h} \right)\left( \frac{\sec^2(t)}{1 - \tan (t)\tan (h)} \right)$$ Having tried various approaches, ...
1
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2answers
217 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 ...
-4
votes
2answers
16 views

Sinusoidal function [on hold]

Any help on the below question would be appreciated. The instantaneous value of current, $i$ amp, at $t$ seconds is given by $i= 15 \sin (100\pi t + 0.6)$ Find: a. the amplitude b. the period c. ...
0
votes
0answers
25 views

How to transform the graph of the sine function [on hold]

Let me ask how to transform the sine function for a given range of the x axis so that: the graph becomes to pass a given desired points anytime y=0; and the graph becomes to pass a given desired ...
0
votes
2answers
41 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
51 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 ...
1
vote
1answer
22 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
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2answers
49 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 ...
-2
votes
2answers
58 views

Prove that $16 \cos 12^\circ ·\cos 24^\circ ·\cos 48^\circ· \cos 96^\circ ·\cos 192^\circ = 1$ [on hold]

Prove that $$16 \cos 12^\circ ·\cos 24^\circ ·\cos 48^\circ· \cos 96^\circ ·\cos 192^\circ = 1$$ Thanks.
0
votes
4answers
103 views

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

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
19 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
votes
2answers
58 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 ...
0
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0answers
55 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
2
votes
2answers
94 views

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

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
votes
0answers
38 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 $ ...
0
votes
1answer
60 views

Sine & Cosine Word Problem [on hold]

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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1answer
42 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
28 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
37 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.
2
votes
2answers
55 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
vote
5answers
57 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
67 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
41 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, $a$, $b$, ...
0
votes
1answer
58 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
votes
4answers
95 views

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

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
0
votes
2answers
54 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.
2
votes
9answers
104 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.
2
votes
4answers
48 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
53 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 ...
0
votes
2answers
38 views

Express $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$ [on hold]

How would I go about expressing $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$, where $A > 0$ and $0 < a < \pi/2$?
0
votes
0answers
31 views

Formal Trigonometric Refrence

I'm Using a textbook for mathematic which is produced to learn for normal students. Here I'm giving the link of chapter of trigonometric functions of my textbook : ...
-3
votes
3answers
41 views

Simplify $\sec(x + \pi/2)$ [on hold]

How would I go about simplifying the equation: $\sec(x + \pi/2)$ I have no idea where to start with this.
0
votes
1answer
37 views

Trigonometry Question - Tough one [on hold]

If in triangle ABC, sin A sin B sin C + cos A cos B = 1. Then find the value of sin C.