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
votes
4answers
17 views

How do you solve for θ in the equation $tan \frac{θ}{5} + \sqrt{3} = 0$

$$\tan \frac{θ}{5} + \sqrt{3} = 0$$ Alright so the $\frac{θ}{5}$ is confusing me. Would it be wrong to do $$\tan \frac{θ}{5} =-\sqrt{3}$$ $$ \frac{θ}{5} =\tan^{-1}(-\sqrt{3})$$ $$ θ = ...
0
votes
0answers
14 views

Arc Length for Superposition of Sinusoidal Curves

I am wanting to compute the arc length, $s$, of a superposition of two sinusoidal functions--say $$y(x) = A\cos\left(k_1 x\right)+B\cos\left(k_2 x\right).$$ There is a special relationship between ...
0
votes
3answers
27 views

Find solutions and solutions in the given interval

So I asked this before with a similar question, and while I got the answer, I still don't understand how to figure out what integer(s) $k$ is. An equation is give (express your answer in terms of k, ...
0
votes
3answers
38 views

$2 \sin 3θ + 1 = 0$ Find solutions and solutions in the given interval

An equation is give (express your answer in terms of k, where k is any integer) $$2 sin 3θ + 1 = 0$$ (a) Find all solutions of the equation. (b) Find the solutions in the interval $[$$0$, $2π$$)$. ...
0
votes
7answers
102 views

Solve the trig equation $\cos\theta − \sin\theta = 1$

Solve the given equation. Let k be any integer. $$\cos θ − \sin θ = 1$$ What do I do? Am I allowed to square everything? I was thinking about squaring everything and then substituting in $1-\sin^2θ$ ...
3
votes
3answers
43 views

Solve the trigonometric equation $\csc^2 \theta= 5 \cot \theta + 7$

Solve the given equation. Let k be any integer. $$\csc^2 θ = 5 \cot θ + 7$$ I just need the first step or two please. I tried converting it: $$\frac{1}{\sin^2 θ} = \frac {5\cosθ}{\sinθ} + 7$$ ...
1
vote
1answer
39 views
-1
votes
3answers
54 views

Trigonometric double identities [on hold]

Prove the following trigonometric identity: $$(\sec (x) - 1)(\sec (x) + 1)=\frac{\sin^2(x)}{\cos^2 (x)}$$
0
votes
0answers
23 views

Hyperbolic functions. [duplicate]

We know the geometric interpretation of sinx, cosx and tanx. But i don't know what is the geometric interpretation of sinh(x), cosh(x) and tanh(x). Please show what is that(i.e side etc) which is ...
1
vote
1answer
23 views

Prove that in every acute triangle, the equation stands: $h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $

Prove that in every acute triangle, the equation stands: $$h_c = c \cfrac{\tan(\alpha)\cdot \tan(\beta)}{\tan(\alpha)+\tan(\beta)} $$
6
votes
2answers
519 views

A problem in integration.

As you know from basic trigonometry that $\sin(2x) = 2\sin(x)\cos(x)$. If you integrate both sides with respect to x, one finds $$\int \sin(2x) \ dx = -\frac{1}{2}\cos(2x)+c$$ on the left hand side ...
-1
votes
0answers
54 views

How to use first derivative of a square root to find $x$ [on hold]

I wanted to differentiate the pythagorean theorem to find the hypotenuse of a triangle, but did not know it's height exact value, which is $(100-x)$. The theorem gave me the equation $$ C=\sqrt{125^2 ...
6
votes
3answers
100 views

When does this equation $\cos(\alpha + \beta) = \cos(\alpha) + \cos(\beta)$ hold?

I come across this problem in an advanced maths textbook for grade 11 in my country. And it's marked a star, which means that it's a difficult exercise, and so, no solution for this problem is given. ...
1
vote
3answers
92 views

The simplification of divided difference of cosine function

What is the following limit? $$\lim_{h \to 0}\frac{\cos(\pi/2+h)-\cos(\pi/2)}{h}$$ Why when simplified do you get $(-\sin(h))/h)$?
3
votes
1answer
29 views

Simplify multinomial trig expansion to fewer linear terms like Wolfram?

I'm mystified as to how Wolfram Alpha's TrigReduce[expr] does its magic, particularly on sums of long products that look like $$A \cos^a(B_1 x) \sin^b(B_2 x) \cos^c (B_3 x) \sin^d (B_4 x) \dots$$ I ...
2
votes
3answers
46 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
-1
votes
0answers
27 views

Trigonometry - Application of Sine and Cos rules [on hold]

Postman Pete leaves the post office and makes deliveries for 2.5 km on Aiden Rd, which is on a bearing of 072 degrees True from his starting point. Pete then heads due north of Sari Boulevard for a ...
0
votes
2answers
44 views

Triangle $ABC$ is isosceles with $AB=AC$, and $D$ is the midpoint of $AB$. If $\angle{BCD}=\angle{BAC}=θ$, then $\cos θ$ equals…?

Triangle $ABC$ is isosceles with $AB=AC$, and $D$ is the midpoint of $AB$. If $\angle{BCD}=\angle{BAC}=θ$, then $\cos θ$ equals...? I was doing some UKMT past paper questions - this question is ...
0
votes
1answer
34 views

Looking for an alternative proof of the angle difference expansion

I have thought about this for a while and have no progress. Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
5
votes
1answer
96 views

When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
3
votes
1answer
68 views

How to find unkown height of triangle without hyptenuse

I been trying to solve this question and have tried to solve it for many days, but do not know how, any help would be much oblidged. A cable company owns the roads marked with the dotted lines in ...
0
votes
2answers
47 views

Solve the equation $(tan θ − 2)(9 sin^2 θ − 1) = 0$

Solve the given equation. (Enter your answers as a comma-separated list. Let $k$ be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) ...
1
vote
3answers
25 views

Write the trigonometric expression as an algebraic expression

Write the trigonometric expression as an algebraic expression. $6 \cos(2 \cos^{-1} x)$ Can someone explain to me how to do this? I tried it on my own after watching a youtube video from patrickjmt ...
3
votes
1answer
64 views

When is the cosine of $\pi/n$ of a certain form?

I have a few questions concerning $\cos(\frac{\pi}{n})$. Are there characterizations for the values $n \in \mathbb{N}$, such that $\cos(\frac{\pi}{n})$ ... is an algebraic number? ... can be written ...
-7
votes
0answers
40 views

How to solve equation like this [on hold]

A trigonometric equation has the following solution: $ x=20°+n*15°$ where n is an integer . which answer(s) of the of the following answers give(s) the exact same result, note that m is also an ...
2
votes
2answers
25 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
3
votes
5answers
56 views

Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
0
votes
2answers
49 views

Trigonometry problem

Okay..this one simple problem but I am really stuck and have no idea how to start.. $\cos(a-b)+\cos(b-c)+\cos(c-a)=-\frac32$ we need to prove $\cos(a)+\cos(b)+\cos(c)=\sin(a)+\sin(b)+\sin(c)=0 $
0
votes
3answers
34 views

If $a \sin \alpha = b \sin \beta$, then show that $ b \cot\alpha + a \cot \beta = (a+b)\cot \frac{\alpha +\beta}{2}$

The question is: If $a \sin \alpha = b \sin \beta$, then show that $$ b \cot\alpha + a \cot \beta = (a+b)\cot \frac{\alpha+\beta}{2} $$ Could I get any hints to the problem? When do I need to ...
-1
votes
4answers
31 views

prove that $\sin^2 b +\sin^2 c - \sin^2 a= -2\cos a \sin b\sin c$. [on hold]

I tried nearly but couldn't prove it afterwards. Please tell me the the way to prove this. The conditions of the problem is that $a+b+c=0$.
-3
votes
0answers
36 views

solve another system of three equations [on hold]

I have: $x=\dfrac{-.5b-.5c+.25d}{b+c+d}$ $y=\dfrac{.5b\sqrt{3}+.5c\sqrt{3}+.25d\sqrt{3}}{b+c+d}$ $z=b+c+2d$ I need help moving the $b$, $c$, and $d$ to the Left-hand-side; and moving the x, y, and ...
0
votes
2answers
52 views

Finding the Zeroes of a Second Derivative to Determine Points of Inflection

I have been told to graph the function $y=\cos \left(x^2\right)$ for $-2π ≤ x ≤ 2π$. I have determined key features of the graph but need help when it comes to determining the points of inflection for ...
0
votes
2answers
37 views

Trigonometric Form of Complex Numbers question. [on hold]

What is the following quotient expressed in polar form: $$\frac{10(\cos(35^{\circ})+ isin(35^{\circ}))}{5(\cos(100^{\circ}) +i\sin(100^{\circ}))}?$$ Please enter your answer in cis notation and ...
3
votes
2answers
92 views

How to prove an identity (Trigonometry Angles--Pi/13)

In this page http://mathworld.wolfram.com/TrigonometryAnglesPi13.html I found equation (11) and (12). $$\cos^2\frac{\pi}{13}+\cos^2\frac{3\pi}{13}+\cos^2\frac{4\pi}{13}=\frac{11+\sqrt{13}}{8}$$ ...
0
votes
0answers
22 views

Smoothly interpolating between functions to create a bouncing wave

How can I create a function which allows me to control the roundness of a wave so I can transition between an Round Wave -> Linear Wave -> Inverted Round wave? I've made a function which creates a ...
2
votes
3answers
83 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Find max of $$S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$$
1
vote
2answers
61 views

Solve $2+ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = 4\sin^2{\frac{x}{4}}$

$$2+ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = 4\sin^2{\frac{x}{4}}$$ My try: $$ \cos{\frac{3x}{2}} + \sqrt{3} \sin{\frac{3x}{2}} = \sqrt{4}\left(\frac{\sqrt{3}}{2} \sin{\frac{3x}{2}} + ...
0
votes
2answers
15 views

In the sin of two angles are equal, then proving that two angles are equal - w.r.t different traingles

From the text book: What do they mean by: Therefore, AC/PR = AB/PQ ? Is the / division or ratio? What rule says that AC/PR = AB/PQ in this example? What do ...
4
votes
2answers
69 views

$D, E, F$ are respectively projection of $O$ on $BC, CA, AB$. Prove that $\cot{\angle ADB} + \cot{\angle BEC} + \cot{\angle CFA} =0$

Let $O$ be an arbitrary point located inside the triangle $ABC$. Let $D, E, F$ be (respectively) the projections of $O$ on $BC, CA, AB$. Prove that $$\cot{\angle ADB} + \cot{\angle BEC} + ...
-3
votes
0answers
26 views

why $\cos 2x$ is positive for $0 < k < 1$? [on hold]

The reflex angle x is such that $\sin x = –k$, where $0 < k < 1$ How to explain why $\cos 2x$ is positive for $0 < k < 1$?
-3
votes
1answer
31 views

find the exact value of $\cos^2x$ and $\csc x$. [on hold]

Given that $x=\tan ^{-1}(\frac{1}{3})$, find the exact value of $\cos^2x$ and $\csc x$. How to find this without using calculator?
0
votes
2answers
56 views

Find the smallest value of $x$ such that $10\cos\left (\frac{x+1}{2}\right)=3$ [on hold]

Given that $x$ is measured in radians and $x > 10$, find the smallest value of $x$ such that $$10\cos \left(\frac{x+1}{2}\right)=3$$ How to solve this question? I've no idea.
1
vote
1answer
36 views

Show that $x^2+y^2$ is constant for all values of $\theta$.

Given that $x=3\sin \theta-2 \cos \theta$ and $y=3\cos \theta+2 \sin \theta$ i)Find the value of the acute angle $\theta$ for which $x=y$ ii)Show that $x^2+y^2$ is constant for all values of ...
4
votes
0answers
33 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
1
vote
1answer
27 views

Express various trig functions in terms of the sine.

The acute angle $x$ radians is such that $\sin x = k$, where $k$ is a positive constant. Express, in terms of $k$. i) $\sin (2\pi-x)$ ii) $\tan(\frac{1}{2}\pi-x)$ iii) $\cos (\pi+x)$ My attempt: ...
1
vote
1answer
35 views

Show that the angle between $OP$ and the normal to the curve at $P$ satisfies the following

I'm struggling to answer the following question below I've already worked out the gradient to the curve at $P$, but I'm having difficulty answering the second part of the question. MY attempt is as ...
3
votes
2answers
59 views

Easy question Find $\sin 2x$, $\cos 2x$, and $\tan 2x$

Ok so I was absent from school yesterday because long story short I had no way to get to class b/c something happened last minute. I'm pretty sure this is easy but I keep getting the wrong answer for ...
-1
votes
3answers
34 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 $\sin 2\beta$ ?
2
votes
4answers
88 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 ...
-1
votes
1answer
53 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.