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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Derivative of $\sin(x)$: how to avoid circular reasoning?

Dad here, helping his kid out with math. I get it that to determine the derivative of $\sin(x)$ you do: $$\frac{\sin(x+h) - \sin(x)}h = \frac{\cos(x)\sin(h)}h \to \cos(x)$$ However I assume that $\sin(h) = h$ for $h\to 0$. This correct because the…
ADBF
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If $x_1, x_2$ are roots of $a\cos x+b\sin x+c=0$ for $x_1+x_2\ne 2k\pi$ show that $\sin(x_1+x_2)=\frac{2ab}{a^2+b^2}.$

If $x_1, x_2$ are roots of $a\cos x+b\sin x+c=0$ for $x_1+x_2≠2kπ$ show that $\sin(x_1+x_2)=\frac{2ab}{a^2+b^2}$ What I've done till now: $$a\cos x_1+b\sin x+c-(a\cos x_2+b\sin x_2+c)=0$$ $a\cos x_1+b\sin x_1-a\cos x_2-b\sin x_2=0$ $a(\cos x_1-\cos…
EL02
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Show that $\sqrt{\frac{1+2\sin\alpha\cos\alpha}{1-2\sin\alpha\cos\alpha}}=\frac{1+\tan\alpha}{\tan\alpha-1}$

Show that $\sqrt{\dfrac{1+2\sin\alpha\cos\alpha}{1-2\sin\alpha\cos\alpha}}=\dfrac{1+\tan\alpha}{\tan\alpha-1}$ if $\alpha\in\left(45^\circ;90^\circ\right)$. We have…
Math Student
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How can I prove that $\sin (10^\circ), \sin(1^\circ), \sin(2^\circ), \sin(3^\circ), \tan(10^\circ)$ are irrational

Prove that $\sin (10^\circ)$, $\sin(1^\circ)$, $\sin(2^\circ)$, $\sin(3^\circ)$, and $\tan(10^\circ)$ are irrational. My Attempt: Let $x = 10^\circ$. Then $$ \begin{align} x &= 10^\circ\\ 3x &= 30^\circ\\ \sin (3x) &= \sin (30^\circ)\\ 3\sin…
juantheron
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Help me solve a trigonometric equation

I am doing some work in RF circuit design. I need to solve an equation for my design: $$\frac 1{\cos(t_1)}+\frac 1{\sin(t_1)} =\frac 1{\cos(t_2)}+\frac 1{\sin(t_2)}$$ (I created a nicely typed image of this equation but setting don't allow me to…
Md Ayatullah Maktoom
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solving for $x$:$\cos^2 3x+\frac{\cos^2 x}{4}=\cos 3x\cos^4 x$

solve for x:$$\cos^2 3x+\frac{\cos^2 x}{4}=\cos 3x\cos^4 x$$ My attempt: completing square: $${(\cos 3x-\frac{\cos x}{2})}^2=\cos 3x\cos x (1-\cos^3 x)$$ or $$\cos x\cos 3x\ge 0$$ also by some basic identities :$$1+\cos 6x+\frac{1+\cos…
Albus Dumbledore
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What's the point of obtuse angle trigonometry?

What is the point of $\sin$ or $\cos$ or $\tan$ of an obtuse angle? Don't we use these functions to find a missing side in a right angle triangle? So why would I use it on an obtuse angle? I have also done a unit circle.
OmarKhaled
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Proving $\tan\frac{4\pi}{11} + 4\sin\frac{\pi}{11} = \sqrt{11}$

In a similar vein as $\tan\frac{3\pi}{11} + 4\sin\frac{2\pi}{11} = \sqrt{11}$ discussed in this question is this identity: $$\tan\frac{4\pi}{11} + 4\sin\frac{\pi}{11} = \sqrt{11}$$ Trying to adopt a method on the same line however lamentably fails.…
Beedassy Lekraj
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Is this derivation I did for the expansion of $\cos (\alpha+\beta)$ correct and how can I get rid of the $\pm$ symbol in it?

I derived the expansion of $\cos (\alpha + \beta)$ as follows, please take a look : Here, $\angle AOB = \alpha$, $\angle BOC = \beta$, $\angle AOC = (\alpha + \beta)$, $a = \cos \alpha$, $b = \sin \alpha$, $m = \cos \beta$, $n = \sin \beta$, $x =…
Rajdeep Sindhu
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What is the meaning of Trigonometry Identities being true for all values?

"Trigonometric identities hold true for all the values of $\theta$". I can't understand this because there are some values which Trigonometric Identities are undefined. Given the Identities: $$ \sec^2\theta-\tan^2\theta=1;…
欲しい未来
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How is $\cos2\theta = \cos^2\theta- \sin^2\theta$?

On Khan Academy in this video at 3:53 minutes in, Khan uses the equation $\cos2\theta = \cos^2\theta- \sin^2\theta$, where can I view the proof for this statement? (He doesn't explain this part in that particular video)
Simon Suh
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$50\cos^2 x + 5\cos x = 6\sin^2 x$, find $\tan x$

$50\cos^2 x + 5\cos x = 6\sin^2 x$ Find $\tan x$ I used $\cos^2 x + \sin^2 x = 1$ to get the equation $$56\cos^2 x + 5\cos x -6 = 0$$ I then solve this to get $\cos x = \dfrac27, -\dfrac38$ Then I used generic trig ratios to get $\tan x =…
max532_
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Proving $\tan^2\frac{\pi}{4n}\cdot\tan^2\frac{3\pi}{4n}\cdot\,\cdots\,\cdot\tan^2\frac{(2n-1)\pi}{4n}=1$

$$\tan^2\frac{\pi}{4n}\cdot\tan^2\frac{3\pi}{4n}\cdot\cdots\cdot\tan^2\frac{(2n-1)\pi}{4n}=1, \quad \forall n \in \Bbb{N}^*$$ original problem image I tried putting $z=\cos(\pi/4)+i\sin(\pi/4)$, and expressing $\tan$ of $\pi/4$, $3\pi/4$ etc. The…
furfur
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A doubt about the argument that $x=\cos\theta$, $y=\sin\theta$ parametrizes the unit circle

There is the question (example) of how the power series definitions of sine and cosine relate to their unit-circle definitions. In many answers (example), the first step is usually something like this: Show that for all $\theta \in \mathbb R$, we…
user693894
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Interpretations of $\sin(x)$

As sin(x) has an infinite number of maximum and minimums, I wondered if $\sin(x)$ could be interpreted in such a way as: $$ax^\infty+bx^{\infty-1}\cdots zx $$Or something. Am I talking nonsense here or is there actually an interpretation of sine…
LazySloth13
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