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.

29665 questions
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$\cos^n x-\sin^n x=1$

For $0 < x < 2\pi$ and positive even $n$, the only solution for $\cos^n x-\sin^n x=1$ is $\pi$. The argument is simple as $0\le\cos^n x, \sin^n x\le1$ and hence $\cos^n x-\sin^n x=1$ iff $\cos^n x=1$ and $\sin^n x=0$. My question is that any nice…
pipi
  • 2,461
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Proving $a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x-\alpha)$

Show that $a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x-\alpha)$ and find the correct phase angle $\alpha$. This is my proof. Let $x$ and $\alpha$ be the the angles in a right triangle with sides $a$, $b$ and $c$, as shown in the figure. Then,…
W. Zhu
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A generalisation of the Pythagorean trigonometric identity

Is there a theorem which states that $\forall n \in \mathbb Z^+\ \ \exists c \in \mathbb R\ \ \forall x \in \mathbb R$ $$\sum_{k=1}^{2n} \sin^{2n}\left(x + \frac{k \pi}{2n}\right) = c$$ (essentially, a generalisation of $\sin^2x + \cos^2x = 1$ to…
Veeno
  • 63
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If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$

If $\displaystyle a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\displaystyle \frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$ $\bf{My\; Try::}$ Using $$\sin(a+\underbrace{b+c}) = \sin a\cdot \cos (b+c)+\cos a\cdot \sin (b+c)$$ $$ = \sin…
juantheron
  • 53,015
6
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Proving that $\sin(54°)\sin(66°) = \sin(48°)\sin(96°)$

I'm trying to prove that $\sin(54°)\sin(66°) = \sin(48°)\sin(96°)$ but I don't really have a way to approach it. Most of what I tried was replacing $\sin(2x)$ with $2\sin(x)\cos(x)$ or changing sines with cosines but none of that has really…
Max
  • 85
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How to find the $\arcsin 2$?

How would I find $\arcsin 2$? I'm helping my little sister with her calculus "pre-test" before classes begin, and I don't remember how to do it in order to explain to her. Help?
Cody
  • 163
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How is it that $\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$ for all angles, even though the derivation holds only for $\cos A\cos B\neq 0$?

How is it that $$\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$$ for any value of $A$, $B$? I have doubts about this since we arrive at this by dividing the numerator and denominator of $$\frac{\sin(A+B)}{\cos(A+B)}$$ by $\cos A \cdot \cos…
Stephen
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I want to find out the angle for the expression $a^3 + b^3 = c^3$.

like in pythagorean theorem angle comes 90 degree for the expression $a^2 + b^2 = c^2$, however I know that no integer solution is possible.
Rahul
  • 544
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Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$

According to my book $$\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ when $x<0$, $y<0$, and $xy>1$. I can't understand one thing out here that when the above mentioned conditions on $x$ and $y$ are followed then the…
Harsh Sharma
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Easier way to show $\cos(20^\circ)\cos(30^\circ)\cos(40^\circ)=\cos^2(10^\circ)\cos(50^\circ)$

The only way I can show $$\cos(20^\circ)\cos(30^\circ)\cos(40^\circ)=\cos^2(10^\circ)\cos(50^\circ)$$ stinks of sweat and brute force: If $\cos(10^\circ)=x$ then using $\cos(a+b) = \cos a \cos b - \sin a \sin b$ and $\cos^2 + \sin^2 = 1$ you…
Mark Fischler
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If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt.

$$ 0<\theta<\pi/2$$ and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$ then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$ I tried to use the above given equation of all the trigonometric ratios,…
Harsh Sharma
  • 2,369
6
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4 answers

Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$

Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$ $\bf{My\; Try::}$ I have solved Using Direct formula:: $$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} =…
juantheron
  • 53,015
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2 answers

Isolating $y$ in $\sin(xy)=\cos(xy)$

Given $\sin(xy)=\cos(xy)$, what is the best way to isolate $y$? Since $\sin(\frac{\pi}{2}) = \cos(\frac{\pi}{2})$ it would seem intuitive to say that $xy=\frac{\pi}{2}$ and thus that $y=\frac{\pi}{2x}$ Is this the correct approach, or am I missing…
Internet Man
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3 answers

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 maybe an alternative exists :)...
jdp89
  • 85
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Prove $\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$ and $\cot{A} + \frac{\sin{A}}{1 + \cos{A}} = \csc{A}$

Can anyone help me solve the following trig equations. $$\frac{\sec{A}+\csc{A}}{\tan{A} + \cot{A}} = \sin{A} + \cos{A}$$ My work thus far…
James
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