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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Solve $\sin(x)\cos(x)=\sin(x)+\cos(x)$

My initial idea was $$(\sin(x)\cos(x))^2=1+2\sin(x)\cos(x)$$ Let $t=\sin(x)\cos(x)$; $$t^2=1+2t \quad\Leftrightarrow\quad t=1-\sqrt2$$ (since $1+\sqrt2>1$). I.e. $$\sin(x)\cos(x)=1-\sqrt2 \quad\Leftrightarrow\quad \tfrac12\sin(2x)=1-\sqrt2…
mf67
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Formula for $a\sin(\theta)+b\cos(\theta)$ confusion

I've been doing some trig lately and in a reference sheet I was given the following formula that I tried to prove $$ a\sin(\theta)+b\cos(\theta)=\frac{a}{|a|}\sqrt{a^2+b^2}\sin(\theta+\alpha) $$ with $\alpha=\arctan\left(\frac{b}{a}\right)$ and…
Anson Pang
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Prove that $\cot 99^\circ=3\cot36^\circ-\sec18^\circ-\csc18^\circ$

Prove that: $$\cot 99^\circ=3\cot36^\circ-\sec18^\circ-\csc18^\circ$$ Actually, I have a proof here (see edit section at the bottom of the text), which I needed to solve a geometry problem fully. It's definitely correct but I don't like it. The…
Saša
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How to determine whether all solutions to $\sin(ax) + \sin(bx) + \sin(cx)=0$ in are rational multiples of $\pi$

I was messing around on Desmos trying to create trigonometry problems when I came across the following: For what positive integers $a,b,c$ is it true that all possible roots of $$\sin(ax)+\sin(bx)+\sin(cx)=0$$ are rational multiples of $\pi?$ By…
x3yukari
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Trigonometry in Triangles Without Right-Angles

Could you please help by showing me how I can find the unknown sides for the triangles below?
jaykirby
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Trigonometric identity: $\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$

I have to prove the following result : $$\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$$ I tried converting $\tan\theta$ & $\cot\theta$ into $\cos\theta$ and $\sin\theta$. That led to a huge…
user76849
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Trigonometric near-identity

The parametrized curve $$ \left( \sec\theta+\csc\theta,\ 2\sqrt{2}\csc(2\theta) \right), \qquad \frac{10}{100} \le\theta\le\frac{142}{100} $$ looks to the naked eye like a straight line. The $y$-intercept is not $0$ and the slope is a number that…
Michael Hardy
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Maximum of a trigonometric function without derivatives

I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or identity? $f(x)=\sin(2x)+2\sin(x)$
EricAm
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Why is $\cos(-x)$ always $\cos(x)$?

While converting angles in trigonometric function I was taught (or I understood) the following We take coordinate axes and a ray which has it’s pivot at the origin. Like so So considering an angle, we rotate the ray in the anti clockwise direction.…
Natru
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solving trigonometry

Solve $16 \cos^2 x + 6 \sin x = 17$ for $0 < x < 2\pi$ Steps: $16(1-\sin^2x)+6\sin x = 17$ $16\sin^2x-6\sin x + 1 = 0$ let $y = \sin x$ $16y^2 - 6y + 1 = 0$ I was not able to solve this quadratic equation. It has no real roots. What was done…
Joe
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How to plot cosine with trigonometric identity

There is the well-known trigonometric identity: $\sin^2(x)+\cos^2(x) = 1$. Without giving a second thought I took use of it to convert from cosine to sine. Especially in physics involving reflections: $\frac{n_2}{n_1}\,\cos(\alpha_2) =…
Leon
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trigonometrical inequality $\tan{A} + \tan{B} + \tan{C} = \tan{A}\tan{B}\tan{C}$, Hint to prove $S2\neq 1$?

I was studying conditional identities for triangle in trigonometry where I had to prove that $\tan{A} + \tan{B} + \tan{C} = \tan{A}\tan{B}\tan{C}$ So I started with $\tan({A+B+C)}=\frac{S1-S3}{1-S2}$ where $S1=\sum_{cyc}\tan A$, $S2=\sum_{cyc}\tan…
Lalit Tolani
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How am I supposed to expand $\sin^2 A + \sin^4 A = 1$ into $1 + \sin^2A = \tan^2A$?

My question is how can i expand $$\sin^2 A + \sin^4 A = 1$$ into: $$1 + \sin^2A = \tan^2A$$ I tried quite a few ways I know but all of them kinda felt random. i am not sure how to share my trials here. I am quite beginner in trigonometry. it is one…
asierx
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Given $\sin{x} = \frac{1 + \sqrt{5}}{4}$, find $x$

If I have been given that $$\sin{x} = \frac{1 + \sqrt{5}}{4}$$ is there any way using which I can find out what the value of $x$ is? Sure I can remember the values, but I was curious whether if I knew the values of $\sin{\frac{\pi}{2}},…
marks_404
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Explanation of where this trig identity comes from

I'm working on a problem but it's been a while since I last saw trig identities so I'd love some help or being pointed in the right direction. Basically, I'd like to understand where this identity comes from; $$\tan(2t) = \dfrac{2\tan(t)}{1 -…
jm22b
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