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

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2
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3answers
58 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
4answers
51 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
42 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 ...
8
votes
1answer
89 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} ...
-2
votes
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
38 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
vote
2answers
212 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
40 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
vote
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
52 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
102 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?
0
votes
3answers
17 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
54 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
votes
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
89 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
vote
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
27 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
54 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
55 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
vote
4answers
66 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
vote
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
57 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
94 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
53 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
103 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.
0
votes
3answers
47 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
44 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..
0
votes
3answers
95 views

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

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
vote
4answers
55 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
27 views

Acute angle and trigonometric functions

Given that $\theta$ is an acute angle and $\cos\theta = \dfrac{7}{25}$. Find: $\tan\theta$, $\sin\theta$, $\sec\theta$.
0
votes
0answers
30 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.
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votes
2answers
43 views

Course And Bearing Word problem. [on hold]

**A plane flew 150 miles on a course of 220 degrees and then 130 miles on a course 130 degrees. Then the plane returned to its starting point via the shortest route possible. Find that shortest ...
3
votes
2answers
57 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
60 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
732 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
37 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 ...
0
votes
2answers
26 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