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
17 views

how to take out oscillations of a sine wave and leave only amplitude?

There is a maths trick to take out the oscillations of a sine signal of varying amplitude, to keep only it's maximum transient amplitude in between peaks. It's something like the the square root of: ...
1
vote
3answers
64 views

why $\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\beta - \alpha)$?

I'm studying linear algebra and there is a chapter in a book that says about unit vector and it says this $$ \cos\alpha \cos\beta + \sin\alpha \sin\beta = \cos(\beta - \alpha) $$ Why?? I'm newbie and ...
0
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0answers
14 views

Relative velocity- Finding the direction of wind.

An aircraft is flying due south at $350~\text{kmh}^{-1}$. The wind is blowing at $70~\text{kmh}^{-1}$ from the direction of $\theta$, where $\theta$ is acute. Given that the pilot is steering the ...
1
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1answer
22 views

Find a right triangle section of a larger one

I'm programming a game, and find myself stumped. I know the target ball (dx, dy) and (cue x, cue y) and the x value of the camera. It's been 20 years and I'm sad to say I've lost some ground here. ...
0
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1answer
19 views

Tangent parallel to the initial line for polar equation =, can r^2 be used instead?

Given a formula for a polar equation: $$\ r^2 = a^2 \cos^22 \theta $$ It could be said that to find the points parallel to the initial line, $$\frac{dy}{dx} = \frac{d (r\sin\theta)}{d\theta} = 0$$ ...
6
votes
4answers
84 views

In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is

In a $\triangle ABC,a^2+b^2+c^2=ac+ab\sqrt3$,then the triangle is $(A)$equilateral $(B)$isosceles $(C)$right angled $(D)$none of these The given condition is $a^2+b^2+c^2=ac+ab\sqrt3$. Using sine ...
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1answer
30 views

Solve:$ 45=-21.9*cos(\frac {\pi}{6}(t-1))+51.6$

I'm interested in learning how to solve this without a graphing calculator. For $t = 0$ to $12$, $$ 45=-21.9\cdot\cos(\frac {\pi}6(t-1))+51.6$$
0
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1answer
112 views

How do I find similar relationships between $x$ and $y$ in the following cases (Trig)? [duplicate]

If $\cos(x)=\cos(y)$, then $x = 2k\pi \pm y$ for $k$ in $\mathbb Z$. How do i find similar relationships between $x$ and $y$ for: a) $\sin (x) = \sin(y)$ b) $\cos (x) = \cos(y)$ and $\sin(x) = \sin ...
2
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2answers
36 views

Show that $\tan \alpha \tan \beta + \tan \beta\tan \gamma +\tan \gamma\tan \alpha =1$

I would appreciate if somebody could help me with the following problem: Q: Show that If $\alpha + \beta + \gamma = {\frac{ \pi}{2}}, \alpha,\beta, \gamma>0 $ then $$\tan \alpha \tan ...
0
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1answer
29 views

How to calculate a point between two angled lines based on distance from the lines?

Please take a look at the picture below for the diagram reference: I am trying to calculate the point where it is perfectly 3.3 cm vertically from the 44.52 cm line AND 5.5 cm horizontally from the ...
0
votes
2answers
35 views

Proof of $\sin (\frac{3 \pi}{2}-A)=-\cos (A)$ geometrically

We already know that $\sin (\frac{3 \pi}{2}-A)=-\cos (A)$ but is there any method to prove it geometrically? Could someone suggest something?
0
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0answers
14 views

How to find the third point of a triangle in a 3D space (arm rig)

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
1
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1answer
26 views

How do I calculate the third point of a triangle in a 3Dimensional Plane

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
0
votes
1answer
41 views

How to find a point equidistant between two points on a sphere.

So I have this problem involving astronomy, but because astronomy uses all sorts of fancy words I'm going to make it more simple by using an analogy of the earth. The process, mathematically would be ...
0
votes
2answers
20 views

When solving for $\sin$ or $\cos(2t)$ given $\sin$ or $\cos(t)$ is the quadrant relevant?

I notice some of the homework problems in my book ask me to find the sine or cosine of twice an angle given the sine or cosine of the angle. They also mention $P(t)$ is in some given quadrant. My ...
2
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0answers
15 views

Question related to trigonometric identities and ratios [duplicate]

Prove that $$\tan \frac{\pi}{30} \tan \frac{7\pi}{30} \tan \frac{11\pi}{30} \tan \frac{13\pi}{30}=1$$ I tried this and i was able to reduce it to $$\cot \frac{\pi}{30} \cot \frac{2\pi}{30} \cot ...
1
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3answers
52 views

Proving a trigonometric identity: [duplicate]

Prove that $\sin \frac{{2\pi }}{7} + \sin \frac{{4\pi }}{7} + \sin \frac{{8\pi }}{7} = \frac{{\sqrt 7 }}{2}$. I have tried to square both side and got ${\sin ^2}\frac{{2\pi }}{7} + {\sin ...
0
votes
0answers
21 views

Infinite number of properties of the sine function graph

Since we can differentiate the sine function an unlimited number of times, does this imply that the graph at every point have an infinite amount of properties like slope, curvature, change in ...
0
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2answers
57 views

When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary?

Sketch the sets on which $\sin\colon\mathbb{C}\to\mathbb{C}$ is purely real/imaginary. My current result is that for purely real numbers $\sin$ is purely real and for purely imaginary numbers ...
0
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0answers
61 views

Did the book make a mistake in the identity$\frac{1}{\cos 2x+\tan x} = \sin 2x$?

EDIT: OP says that they misread the text from which this question was drawn. It actually said $$\frac{1}{\cot(2x) + \tan(x)} = \sin(2x)$$ where $\cos$ has been replaced by $\cot$; that identity is ...
2
votes
2answers
26 views

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$

show that $y = \cos x$, has a maximum turning point at $(0, 1)$ and a minimum turning point at $(\pi, -1)$ Turning points occur when the gradient is 0 or $\frac{dy}{dx} = 0$ $f(x) = \cos x$ ...
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1answer
44 views

Interpreting a “sunrise function”: $t=-1.4\sin\left(\frac{2\pi}{365} (d-75)\right) + 7$ [closed]

In Africa, the time for sunrise any day during the year can be shown as the formula: $$t=-1.4\sin\left(\frac{2\pi}{365} (d-75)\right) + 7$$ where $t$ is the time in hours after midnight and ...
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1answer
63 views

The value of $\tan \frac \pi {-12}$ [closed]

Use a compound angle formula to determine the exact value of $\tan \frac \pi {-12}$.
0
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2answers
37 views

Finding the max/min turning points of $y=\sin x$

Use differentiation to show that $$y = \sin x$$ has a maximum turning point at $\left(\frac{\pi}2, 1\right)$ and a minimum turning point of $\left(\frac{3\pi}2, -1\right)$. I know that the ...
1
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2answers
56 views

Evaluate $\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$ without L'Hopital or Taylor series

I want to evaluate $$\lim _{x\to \pi }\left(\frac{\sin x}{\pi ^2-x^2}\right)$$ without using L'Hopital's rule or Taylor series. My thinking process was something like this: in order to get rid of ...
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votes
2answers
44 views

TRIG: Prove this identity [closed]

$$\dfrac{\sin(2a)}{2+2\cos(2a)} = \dfrac{\sec^2 a-1}{2\tan a}$$ Please advise on how to prove this identity so that the left hand side $=$ right hand side. Thank you in advance.
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2answers
36 views

TRIG: Amplitude and Period of a Cosine Function / Identity [closed]

Determine algebraically the amplitude and period of the following function: $$y=\cos\left(\frac{3\pi}{4} + \theta\right) + \cos\left(\frac{3\pi}{4} - \theta\right)$$ Can someone please explain step ...
3
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4answers
58 views

value of $\tan(A)$

What is value of $tanA$ if $2\tan(2A)+4\tan(4A)+\frac{8}{\tan(8A)}=0$ writing everything in $\tan(A)$ and solving for $t$ is next to impossible without maths engines. So i am seeking for a shorter ...
1
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1answer
47 views

Solve the trigonometric equation $2 \alpha + \sin(\alpha \pi) - \alpha \cos(\alpha \pi)=0$

How we can solve in $\mathbb{R}$ the following equation? $$ 2 \alpha + \sin(\alpha \pi) - \alpha \cos(\alpha \pi)=0 $$ I'm lost.
2
votes
3answers
51 views

Doubt in solving $\sec^{-1}\sqrt{5}+\csc^{-1}\frac{\sqrt{10}}{3}+\cot^{-1}\frac{1}{x}=\pi$

Find the value of $x$ if $$\sec^{-1}\sqrt{5}+\csc^{-1}\frac{\sqrt{10}}{3}+\cot^{-1}\frac{1}{x}=\pi$$ First i tried to calculate the value of ...
0
votes
1answer
14 views

Trigonometric function with a theoretical scenario used to find missing variables?

For my pre-calculus class I am given a theoretical scenario and I am tasked with finding the different time(s) an object within the scenario will be "x" inches above equilibrium. The prompt given to ...
1
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1answer
26 views

Finding the missing values in a trigonometric theoretical scenario? Rocket launch?

For my pre-calculus class, I am tasked with sketching a diagram illustrating a scenario as well as determining the angle of elevation after the object is "x" meters above the ground. The prompt is: ...
0
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2answers
64 views

What is the value of $\tan (\frac{\pi}{2} - \epsilon)$? [closed]

$$\tan(\frac{\pi}{2}-\epsilon)$$ By epsilon I mean an infinitesimal.
0
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0answers
14 views

Solving Non-Linear Simultaneous Equations

Equation 1: $$ L_{5}^{2} = (C_0 - C_2)^2 + (C_1 \cdot \sin(\theta_2) - C_3 \cdot \sin(\theta_1))^2 + (C_1 \cdot \cos(\theta_2) - C_3 \cos(\theta_1)))^2 $$ Equation 2: $$ L_{7}^{2} = (\lambda \cdot ...
-1
votes
2answers
55 views

Help me simplify: $\cos(−\theta) + \tan(−\theta) \sin(−\theta)$ [closed]

Simplify $$\cos(−\theta) + \tan(−\theta) \sin(−\theta)$$ to one term with no negative thetas.
6
votes
3answers
87 views

How to solve $A\tan\theta-B\sin\theta=1$

I was wondering if it is possible to solve $$A\tan\theta-B\sin\theta=1$$ for $\theta$, where $A>0,B>0$ are real constants. For sure this can be straightforwardly implemented numerically, but ...
0
votes
2answers
56 views

Integrate $\sin(3x)\cos(3x)$

Integrate $\sin(3x)\cos(3x)$ I looked at various answers on different sites but still do not understand how to use the u-substitution method in this question or the double angle rule.
0
votes
3answers
49 views

Intersecting circles and the sine and cosine rules

So I wrote a question using three numbers $r_1$, $r_2$ and $l$. I am struggling to solve it "in general" while playing by certain rules. The rules are: no calculator and no half- or double-angle ...
1
vote
1answer
20 views

Variable Separation - Find $\theta$ in $\frac{\cos\left(\frac{\pi}2\cos\theta\right)}{\sin\theta} = \frac{0.8912r}{60I}$

Could anyone kindly help me out to obtain an expression for θ in the given equation - $$\frac{\cos\left(\frac{\pi}2\cos\theta\right)}{\sin\theta} = \frac{0.8912r}{60I}$$ I can't separate $\theta$ ...
4
votes
2answers
404 views

Forcing Bijectivity

I'm working out of the Nakahara text in mathematical physics, and I'm presented with a map $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $ f:x \mapsto \sin(x) $, and told that it is neither ...
1
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1answer
45 views

Trigonometry related problem

A trig problem. I cannot solve for the distances: $DP_{1,3}$ and $BP_{2,3}$. I tried law of sines for the triangle $O_2O_4P_{1,3}$ but don' know the above distances and also don't know another ...
1
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1answer
39 views

Solve $\cos 2x - 3\sin x - 1 = 0$ using addition formula

Solve $\cos 2x - 3\sin x - 1 = 0, \quad 0^{\circ} \le x \le 360^{\circ}$ \begin{align} \cos 2x - 3\sin x - 1 = 0 &\iff 1 - 2\sin^2 x - 3\sin x - 1 = 0 \\ &\iff- 2\sin^2 x - 3\sin x = 0 ...
1
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2answers
36 views

Multiple Angle Identity with $\sin(11x)$

I have a problem that I cannot seem to solve. I need to prove the following identity, $$\sin(11x) = 2\sin(8x)\cos(3x)-\sin(5x)$$ I do not understand how to go about this problem, as clearly ...
2
votes
2answers
65 views

What's the point of “trigonometric proofs/identities” in introductory calculus/pre-calculus?

I remember back in high school at some point delving into worksheet after worksheet of trigonometric "identities", the vast majority of which are basically restatements of $\sin^2(x) + \cos^2(x) = 1$ ...
3
votes
1answer
530 views

Why is the double angle formula not used in this solution?

solve $\sin 2x = 0.5$ I am looking at the answer to this question and it does not use the double angle formula to make this: $\sin 2x = 2\sin x \cos x$ And instead use $2x = \arcsin 0.5$ Why is ...
0
votes
1answer
32 views

solve $\cos 2x - \sin x = 0$ using double angle formula

Solve $\cos 2x - \sin x = 0$. \begin{align}\cos 2x = 1 - 2\sin^2 x &\iff 1 -2 \sin^2 x - \sin x = 0 \\ &\iff -2 \sin^2 x - \sin x + 1 = 0 \\ &\iff 2 \sin^2 x + \sin x - 1 = 0 \\ ...
0
votes
1answer
16 views

Jacobian of a system of equations

I'm asked to compute the Jacobian of a system of equations $x_1^4+x_2^4-1=0$ $x_2-\sin(5x_1)=0$ $x_1-x_3^2=0$ What's the Jacobian of a system of equations? Do I perhaps need to infer the individual ...
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votes
1answer
55 views

The lateral surface area of a right circular cone is three times the surface area of its inscribed sphere. Find the vertex angle (cone)

I am quite confused on how to attack this problem... It's not homework I tried to use the equations of the surface area of a sphere and the lateral area of the cone to find the answer, but it didn't ...
-1
votes
0answers
13 views

Plot current-voltage curve where $\langle \dot{\theta} \rangle$ is dependent on $I$

I have been given the following question as part of an investigation for my Applied Maths course: "[...] Then construct the current-voltage curve, i.e. the dependence of $\left< \dot{\theta} ...
1
vote
1answer
50 views

Find the second focus of an ellipse given one focus, the point furthest from it, an arbitrary other point on the ellipse [duplicate]

I have the following three points: A: One focus of an ellipse C: The furthest point from this focus (the far side of the major axis) D: Some other arbitrary point on the ellipse From there, it's ...