Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [trigonometry]

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

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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How to prove the following identities?

Prove: \begin{align} \tan(A) + \cot(A) & = 2 \text{cosec}(2A)\\ \tan(45^{\circ}+A^{\circ}) - \tan(45^{\circ}-A^{\circ}) & = 2 \tan(2A^{\circ})\\ \text{cosec}(2A) + \cot(2A) & = \cot(A) \end{align} I have got all the formulas that I need…
Sophia
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using trigonometric functions to adjust values to be from -1 to 1

I have some code which is projecting 3D points onto a 2D surface based on the user's viewpoint. The top graph shows the actual points that I get from my code. The y values seem to be correct, but the x values are off. The blue x values in the top…
HappyEngineer
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Fitting data with Cosine

So I'm trying to wrap my head about Graphing Cos and Sin but I have several questions about graphing. I know that the Formula is $$Y= A \sin (t \pm h) + K$$ where $A$ is Amplitude , $t$ is Period, $h$ is horizontal shift, $k$ is the mid line. I…
Cris
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Prove that for $\cos (\alpha ) = \frac{1}{3}$, $\alpha < \frac{\pi}{2} - \frac{1}{3}$

I have the following question in a mock exam: $\beta = \frac{\pi}{2} - \alpha$, show that $\beta > \frac{1}{3}$ From the earlier part of the question we know that $\alpha$ is an angle between two vectors (so it's between $0$ and $\pi$) and that…
Robert
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trigonometric equations calculations and their implementation in practise.

$$\sin (11x)\sin (x) = \cos (10x) $$ How can one solve this ? The formulas for summation or doubling dont seem to be of much help here.
Bak1139
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How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$

Prove the following equation. \begin{eqnarray} \\\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1\\ \end{eqnarray} I can't prove it by many methods I use. Please give me some hints. Thank you for your attention
Casper
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Does $\cos(x+y)=\cos x + \cos y$?

Find the value using a calculator: $\cos 75°$ At first I thought all I need is to separate the simpler known values like this: $\cos 75^\circ = \cos 30°+\cos45° = {\sqrt3}/{2} + {\sqrt2}/{2} $ $= {(\sqrt3+\sqrt2)}/{2} $ This is my answer which…
Sylvester
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from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$

How can you go from $1-\sin x $ to $2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$? I mean how to prove that $1-\sin x = 2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)$?
Machiaweliczny
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Finding a trigonometric function for a problem

The question: At Dolphin Bay the depth of the water at the end of the jetty is 6 metres at high and 4 metres at low tide. High tide occurs at 11am and low tide occurs at 5pm. a). Using the information given find a trigonometric function which models…
Dale Townsend
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Solutions to an underdamped harmonic oscillator equation

This is actually a physics question, but the final part of the solution is pure mathematics. I have wracked my brains trying to figure out a more elegant solution, which I know to exist. There are three points that we know of: $$\begin{align} x(0) &…
Kayle of the Creeks
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Range of a sum of sine waves

Suppose I'm given a function f(x) = sin(Ax + B) + sin(Cx + D) is there a simple (or, perhaps, not-so-simple) way to compute the range of this function? My goal is ultimately to construct a function g(x, S, T) that maps f to the range [S, T]. My…
Zach
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Solving $\cos(x/2) = \sin x$

Can anyone give me just a hint as to how to go about solving this? Yes it is for homework, but I don't want an answer, just a bit of guidance. I'm thinking the half angle formula for the first bit, but it doesn't seem to look…
Josh
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For which real numbers $x$ is $\arcsin(2x^2-1)-2\arcsin x=-\dfrac{\pi}{2}$

$$\arcsin(2x^2-1)-2\arcsin x=-\dfrac{\pi}{2}$$ Thanks in advance
Stanson
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Show that $\cos \pi = -1$

How can I proof that $\cos \pi = -1$. I know that this is the answer if I type it in my calculator. If I draw the unit cirlce, then the answer is also clear, but is there an more mathematical way to show that $\cos \pi = -1$ ? To add some more…
90intuition
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Prove $2\cos^2(x)=1+\cos(2x)$

I need help to prove that: $2\cos^2(x)=1+\cos(2x)$. I know that $\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$, but I don't know how to get to this step without memorizing this, can you possibly draw some sort of triangle to…
Ryan
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