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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Solving $\cos^2 \theta + \cos \theta = 2$

Solve the following for $\theta$: $\cos^2 \theta + \cos \theta = 2$ [Hint: There is only one solution.] I started this out by changing $\cos^2\theta$ to $\dfrac{1+\cos(2\theta)}{2}+\cos\theta=2$ $1+\cos(2\theta)$ turns into…
Austin Broussard
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If $P=\tan(3^{n+1}\theta)-\tan\theta$ and $Q=\sum_{r=0}^n\frac{\sin(3^r\theta)}{\cos(3^{r+1}\theta)}$,

If $P=\tan(3^{n+1}\theta)-\tan\theta$ and $Q=\sum_{r=0}^n\frac{\sin(3^r\theta)}{\cos(3^{r+1}\theta)}$,then relate $P$ and…
Brahmagupta
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Solving $\sin(5x) = \sin(x)$

If I have an equation: $$\sin(5x) = \sin(x)$$ In what case can I equate $$5x = x$$ Is it only when there is a multiply of $2\pi n$ on either side, where n is any integer so $$ 5x = x+2\pi n$$ Also with this method can I get every possible solution…
bigfocalchord
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Trigonometry conversion rules, why this way?

We have domain $\,[0, 2\pi]\,$ and the following functions are given: $$f(x)=\cos(2x) \text{ and } g(x)=\sin(x-\pi/3)$$Solve exactly: $\,f(x)=g(x)$ Why does one solve: (right way) $$\cos(2x)=\sin(x-\pi/3)\\\cos(2x)=\cos(\pi/2-(x-\pi/3))\\…
Cro-Magnon
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$x^{2}+2x \cdot \cos b \cdot \cos c+\cos^2{b}+\cos^2{c}-1=0$

Solve this equation : $$x^{2}+2x \cdot \cos b \cdot \cos c+\cos^2{b}+\cos^2{c}-1=0$$ Such that $a+b+c=\pi$ I don't have any idea. I can't try anything.
user315918
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How can trigonometric functions be negative?

I cannot understand why $\cos(180-\theta)$ say is $-\cos\theta$. This is probably because my teacher first introduced trigonometry in triangles. I do not understand it for obtuse angles because I cannot think of them in a right triangle. I…
The Cryptic Cat
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How Were Sine, Cosine and Tangent Formulas Conceived?

I am looking mostly, for a visual answer since I am not so advanced in mathematics. When solving for the opposite side, for example, how does the formula convert degrees to a measurement, say 2 inches? I have read about certain "tables" being used…
user318108
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How to calculate $\sum_{k=1}^{\infty}\operatorname{arccot}\frac{1-k^2+k^4}{2k}$

How to calculate this trigonometric sum? $$\sum_{k=1}^{\infty}\operatorname{arccot}\frac{1-k^2+k^4}{2k}$$
FMath
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Height and Distance problems

A ladder rests against a wall at an angle $\alpha$ to the horizontal. When its foot is pulled away from the wall through a distance $a$, it slides a distance $b$ down the wall and makes an angle $\beta$ with the horizontal. Prove that…
Iaamuser user
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Solving a trigonometric equation

I'm solving this equation: $$\sin(3x) = 0$$ The angle is equal to 0, therefore: $$3x=0+2k\pi \space\vee\space3x= (\pi-0)+2k\pi$$ $$x = \frac {2}{3}k\pi \space \vee \space x = \frac {\pi + 2k\pi}{3}$$ Though, the answer is $$x = k\frac…
Cesare
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Fundamental Theorem of Algebra for Trigonometry

The Fundamental Theorem of Algebra states that "Any polynomial of degree $n$ ... has $n$ roots." Is there anything analogous for trigonometric equations? I've been solving some trigonometric equations, and solving some of the slightly more complex…
StopReadingThisUsername
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Is is possible to simplify the expression $\arctan(y)-\arctan(x)=c$

Is is possible to simplify the expression $\arctan(y)-\arctan(x)=c$. I tried writing the expression in the form $\frac{\arcsin(y)}{\arccos(y)}-\frac{\arcsin(x)}{\arccos(x)}=c$ but it does not lead to anyting. How do I reduce it I want to eliminate…
Santosh kumar Bhal
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How is $\sqrt {2+\sqrt {2+\sqrt {2+}}} ... n $ times = $2\cos( π/2^{n+1})$?

How is $\sqrt {2+\sqrt {2+\sqrt {2+}}} ... n $ times = $2\cos( π/2^{n+1})$? No idea. Please help. I found this identity in a solution of a problem related to limits. Also if any more identities like this then please let me know in the answer…
Akshay Pratap Singh
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Solving $\sin 7\phi+\cos 3\phi=0$

The question is find the general solution of this equation:$$\sin(7\phi)+\cos(3\phi)=0$$ I tried to use the "Sum-to-Product" formula, but found it only suitable for $\sin(a)\pm \sin(b)$ or $\cos(a)\pm \cos(b)$. So I tried to expand $\sin 7\phi$…
Vic.
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Help Verifying Trigonometric Identity

I could really use help, hint or otherwise, in proving a trigonometric identity: We are only allowed to work on one side of the equation. $$\dfrac{2\sin^2(x)-5\sin(x)+2}{\sin(x)-2} = 2\sin(x)-1$$
Daniel B.
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