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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Expressing $\cos\theta - \sqrt{3}\sin\theta = r\sin(\theta - \alpha)$

My book explains that $a\cos\theta + b\sin\theta$ is a sine (or cosine) graph with a particular amplitude/shift (i.e. $r\sin(\theta + \alpha)$) and shows me some steps to solve for $r$ and $\alpha$: $$r\sin(\theta + \alpha) \equiv a\cos\theta +…
PeteUK
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Solve $\sin 5x = \sin x$.

I have to solve the following equation: $$\sin 5x = \sin x$$ This is what I tried: $$5x = x + 2k \pi \hspace{5cm} 5x = \pi - x + 2k \pi$$ $$4x = 2k \pi \hspace{5cm} 6x = (2k + 1) \pi$$ $$\hspace{2cm} x = k \dfrac{\pi}{2}, k\in \mathbb{Z}…
user592938
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which is larger: $\sin(4^\circ)$ or $2\sin(2^\circ)$?

I had two expression that simplified to the ones in the title. Obviously, I can't use a calculator. We didn't learn the double angle (or half angle) formulas so Ill have to find a different way, maybe with the unit circle.
aradarbel10
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Where are secant, cosecant, and cotangent applied?

The textbook we use says: The cosecant function and the secant function are the reciprocal functions of the sine function and the cosine function, respectively, and thus are also periodic functions. Their graphs are not as useful and are…
colboynik
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Show that the general value of $\theta$ satisfying $\sin\theta=\sin\alpha$ and $\cos\theta = \cos\alpha$ is given by $\theta = 2n\pi + \alpha$

The general value of $\theta$ simultaneously satisfying equations, $$\sin\theta = \sin\alpha \quad\text{and}\quad \cos\theta = \cos\alpha$$ is given by $$\theta = 2n\pi + \alpha \ \forall \ n \in \mathbb{Z} \\$$ My attempt: Adding the two…
Simran
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$\cos{(A-2B)}+\cos{(B-2C)}+\cos{(C-2A)}=\cos{(2A-B)}+\cos{(2B-C)}+\cos{(2C-A)}$

Let $A,B,C\in \mathbb{R}$ with $\sin{A}+\sin{B}+\sin{C}=0$. Prove that $$\cos{(A-2B)}+\cos{(B-2C)}+\cos{(C-2A)}=\cos{(2A-B)}+\cos{(2B-C)}+\cos{(2C-A)}$$
math110
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Cosine related to Bitwise XOR through Repeated Roots

I have been doing a lot of calculations involving cosine and have noticed a pattern. I spent some time and developed this algorithm for calculating cosines. To calculate the $\cos(x)$ where $x$ is in radians: Sanitize the input so that $x$ is…
TheNewGuy
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Is there a name for sec(x)'s relationship with tan(x)?

In a couple of trig identities, esp to do with integrals and derivatives, you see a relationship between tan(x) and sec(x). Similarly between csc(x) and cot(x). $ \frac{d}{dx}\tan(x) = \sec^2(x) $ $ \frac{d}{dx}\sec(x) = \sec(x) \tan(x) $ $ tan^2(x)…
bobobobo
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Proving the identity $\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$ by considering a more general form

I came across this exercise: Prove that $$\tan x+2\tan2x+4\tan4x+8\cot8x=\cot x$$ Proving this seems tedious but doable, I think, by exploiting double angle identities several times, and presumably several terms on the left hand side would…
user170231
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Method to find $\sin (2\pi/7)$

I just thought a way to find $\sin\frac{2π}{7}$. Considering the equation $x^7=1$ $⇒(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$ $⇒(x-1)[(x+\frac1 x)^3+(x+\frac1 x)^2-2(x+\frac1 x)-1]=0$ We can then get the 7 solutions of x, but the steps will be very…
JSCB
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Proving the angle sum and difference identities for sine and cosine without involving the functions' geometric meanings

For well known identities $$ \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta $$ $$ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta $$ is it possible to provide a proof which does not involve geometric…
scrutari
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If $\tan 2\alpha \cdot \tan \alpha = 1$, then what is $\alpha$? Different methods give different answers.

If $\tan 2\alpha\cdot\tan \alpha = 1$, then what is $\alpha$? I tried two methods but got two different answers. Method 1: $$\begin{align} \tan 2\alpha\cdot\tan \alpha = 1 &\implies \frac{2\tan \alpha}{1 - \tan ^2 \alpha}\;\tan \alpha = 1…
Adnan Toky
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Prove that $\frac{\sin x}{x}=(\cos\frac{x}{2}) (\cos\frac{x}{4}) (\cos \frac{x}{8})...$

How do I prove this identity: $$\frac{\sin x}{x}=\left(\cos\frac{x}{2}\right) \left(\cos\frac{x}{4}\right) \left(\cos \frac{x}{8}\right)...$$ My idea is to let $$y=\frac{\sin x}{x}$$ and $$xy=\sin x$$ Then use the double angle identity $\sin…
Loo Soo Yong
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How to solve $\sin^2(x)+\sin2x+2\cos^2(x)=0$

How do you solve $\sin^2(x)+\sin2x+2\cos^2(x)=0$? I have been able to rewrite it as $(\sin(x)+\cos(x))^2+\cos^2(x)=0$. Not obviously useful, I think
James Warthington
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Triangle with $\sin^2 A +\sin^2 B =\sin C$

I want to know the triangles which satisfies the following equation : $\sin^2 A + \sin^2 B = \sin C$. Here $A$, $B$, $C$ are angles of a triangle. If we let $a$, $b$, $c$ to be the lengths of edges corresponded to angles, then we multiply $(abc)^2$…
HK Lee
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