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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How to solve $\sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6^\circ$

Question: $ \sin78^\circ-\sin66^\circ-\sin42^\circ+\sin6° $ I have partially solved this:- $$ \sin78^\circ-\sin42^\circ +\sin6^\circ-\sin66^\circ $$ $$ 2\cos\left(\frac{78^\circ+42^\circ}{2}\right) \sin\left(\frac{78^\circ-42^\circ}{2}\right) +…
Abhishek Mhatre
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How does $\cos(\pi ) = -1$?

I know this is a very elementary question, but how does $\cos(\pi) = -1$? I thought the cosine function required a minimum of 2 numbers, the adjacent side and hypotenuse of a triangle?
subtlearray
  • 285
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Sine of an obtuse angle

In the figure above, $\angle MOP=\theta , \angle POP'=90^o$ $$\sin (90^o+\theta)=\sin \angle MOP'=\frac{M'P'}{OP'}(\text{how?})=\frac{OM}{OP}=\cos \theta$$ Sine is opposite side/hypotenuse, then in this case how does the author determine the sine…
Vikram
  • 5,580
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Why this this approach to solving a trig equation yield invalid solutions alongside valid ones?

I've got the following equation: $$\csc(4x) - \cot(4x) = 1$$ $$0 < x < 2\pi$$ At first, I tried solving it like this: $$\frac{1}{\sin(4x)} - \frac{\cos(4x)}{\sin(4x)} = 1$$ $$1 - \cos(4x) = \sin(4x)$$ $$1 = \sin(4x) + \cos(4x)$$ $$1^2 = \sin^2(4x) +…
Eddard Stark
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Showing $\cos (\arcsin(\cos(\theta))) = \lvert \sin (\theta) \rvert$

Where does an absolute function should appear and why? Having the following equation: $$ \cos (\arcsin(\cos(\theta))) = \lvert \sin (\theta) \rvert $$ With $ -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$. I draw a triangle and easily found the…
Dor
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Solve $A = (B+C)\cdot \sin(2\cdot \tan^{-1}(C/D))$ for $C$ algebraically

I came across this equation while working out how to hang a rising gate by offsetting one hinge. $$A = (B + C) \cdot \sin(2 \cdot \tan^{-1}(C / D))$$ I know A, B and D and have to find C. Having no idea how to approach an algebraic solution, I…
emrys57
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$\sin x + c_1 = \cos x + c_2$

While working a physics problem I ran into a seemingly simple trig equation I couldn't solve. I'm curious if anyone knows a way to solve the equation: $\sin(x)+c_1 = \cos(x)+c_2$ (where $c_1$ and $c_2$ are constants) for $x$ without using Newton's…
Nick
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Solve $\cos3x=\cos4x$

I want to solve the equation $\cos3x=\cos4x$. The given solutions are $x= 0$, $2\pi/7$, $4\pi/7$ and $6\pi/7$. My first approach was to write the whole thing in terms of $\cos x$ this gave, $0=(\cos x - 1)(8\cos^3x + 4\cos^2x - 4\cos x - 1)$. This…
Grace
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Could someone please explain double-angle identities?

I don't understand how to do maths, mostly because I don't understand why formulae work they way they do, or the reasoning behind equations, etc. I tried to explain the $\sin(2\theta)$ double-angle identity to myself but failed: Hypothetically if:…
Vesta
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Given $\csc\theta=-\frac53$ and $\pi<\theta<\frac32\pi$, evaluate sine ,cosine, and tangent of $2\theta$

If $\csc\theta=\frac{-5}{3}$, what is the exact value of $\tan(2\theta)$, $\sin(2\theta)$, and $\cos(2\theta)$ on the interval of $\left(\pi, \frac{3\pi}{2}\right)$? I think I'm getting the fraction negatives wrong. I've used the sine…
user242559
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Show that $\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x)$

The question asks to prove that - $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x} = \frac 12 (\tan 27x - \tan x) $$ I tried combining the first two or the last two fractions on the L.H.S to allow me to use the…
MayankJain
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find all possible solutions

The set of all $x$ in the interval $[0,\pi]$ for which $2\sin^2x-3\sin x+1 \geq 0$, is _________________. I have tried by factoring it first and then comparing it with the inequality. My final step was $(\sin x-1)(2\sin x-1) \geq 0$.
Kaushik
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How to determine negative values of arc functions?

Can someone explain me how do they work? I memorized the table of values for sin/cos/tan/cot and radian-degreee values. But, how come, when it is $\arcsin -1/2$ this is $-\pi/6$. So the same thing as $\arcsin 1/2$ just with a different sign. Why it…
Javid Asgarov
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Solving the ArcTan of an angle (Radians) by hand?

How do you solve $\arctan(n)$ to radians by hand? I. e. $\arctan(1)$ >> process >> $\pi/4$ I have this Taylor expansion that allows me to calculate an approximate value for arctan, but am wondering if there's a closed-form solution (Or a more…
Mr. Demetrius Michael
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Trying to prove a trigonometric identity

I've been trying to solve it for quite some time but I still don't get it why it is true. The original equation is: \begin{equation*} 1-\frac{\sin{^2}\theta}{1-\cos\theta}=-\cos\theta. \end{equation*} My work so…
Dormin
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