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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Solving $x = \tan x$

Out of curiosity, I tried to solve the eqation $$x = \tan x$$ but it was harder than I first thought. Eventually I built an algrothim to solve this eqution using the bisection method. But, is there any way to arrive to an exact solutions? I tried…
Ofek Tevet
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How to solve $\sin 2x \sin x+(\cos x)^2 = \sin 5x \sin 4x+(\cos 4x)^2$?

How to solve $\sin 2x \sin x+(\cos x)^2 = \sin 5x \sin 4x+(\cos 4x)^2$? \begin{align*} 2\cos x(\sin x)^2+(\cos x)^2 & = \frac{1}{2}(\cos x- \cos 9x) +(\cos 4x)^2\\ 4\cos x(1-(\cos x)^2)+2(\cos x)^2 & = \cos x + \cos 9x +(2(\cos 4x)^2-1)+1\\ \cos…
Sgg8
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Trying to find $\tan^{-1}x-\tan^{-1}y$ for $\forall~x,y$

Suppose we want to calculate $\tan^{-1}x-\tan^{-1}y$ for $\forall~x,y$ We already know $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\dfrac{x-y}{1+xy}$ for $x>0$ and $y>0$, but we will not make use of it as we have to prove for $\forall$…
user3290550
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Equation $(8\cos^3x+1)^3=162\cos x-27$

Solve equation $$(8\cos^3x+1)^3=162\cos x-27$$ I saw this equation before 5 month, and I couldn't solve it. This isn't homework, etc. (I don't do stuff like this anymore). I am just curious.
Cortizol
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prove that for every parallelogram

Prove that for every parallelogram $$d_1^2+d_2^2=2(a^2+b^2)$$ My attempt is: $d_1^2=a^2+b^2-2ab\cos\alpha$ $d_2^2=a^2+b^2-2ab\cos(180-\alpha)=a^2+b^2+2ab\cos\alpha$ It follows that: $$d_1^2+d_2^2=2(a^2+b^2)$$ but I don't know whether that is…
betmen 12
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Problem involving slope of a plane

I want to solve parts (ii) and (iii) of this problem without using vectors In the region of 3 fixed buoys A, B and C at sea there is a plane stratum of oil-bearing rock. The depths of the rock below A, B and C are 900m, 800m and 1,000m…
Steblo
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What is the domain of the function $\tan\theta_{1}(\theta)=\frac{\sin\theta}{\cos\theta+1}$ and what is $\theta_1^{\max}$?

I have some questions about the lecture that I took today on Physics. Consider the cosine function defined below, $\cos\theta=-\frac{M_2}{M_1}$ $M_1$: Mass of the first object. $M_2$: Mass of the second object. (Sorry for the physical terms, I was…
M. Çağlar TUFAN
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Solve $\sin(12º)\sin(24º)\sin(84º-xº) = \sin(30º)\sin(30º)\sin(xº)$

I'm trying to solve this trigonometric equation: $$\sin(12º)\sin(24º)\sin(84º-xº) = \sin(30º)\sin(30º)\sin(xº)$$ I got here after applying Trigonometric Ceva Theorem. Here, I don't know how to solve it, I tried to use…
Trobeli
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Solving $a \sin(\alpha) - c \sin^2(\alpha) = b \cos(\alpha) - c \cos^2(\alpha)$

$a, b, c$ are given positive integers. I need $\sin(\alpha)$ or $\cos$ or anything simple with $\alpha$ from the equation: $$a \sin(\alpha) - c \sin^2(\alpha) = b \cos(\alpha) - c \cos^2(\alpha)$$
Ivan
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A new method to solve a SAS triangle: $\tan B =\frac{AC \sin C}{BC- AC \cos C}$. Should I publish?

I've discovered a new method to solve SAS triangles without using the law of cosines: In $\triangle ABC$, if sides $AC$ and $BC$, and angle $C$, are known, then: $$\tan B =\frac{AC \sin C}{BC- AC \cos C}$$ Putting in mind that I'm still a…
Ahmed M. Elsonbaty
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Verify the trig identity $\frac{\cot^2\theta-1}{\csc^2\theta}=\csc \theta -1$

I've stumbled across a brain-teaser. After using some identities, I get the left hand side equal to $\cos^2\theta - \sin^2\theta$. I'm not aware of any other identities that could get me to the right hand side. I'm actually leaning towards there…
p3ngu1n
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Solving $\tan 2x=1+2\sin 4x$

Solve: $$\tan 2x=1+2\sin 4x$$ My work: $$\left(\frac{\sin2x}{\cos2x}\right)(1+2\cos2x)(1-2\cos2x)=1$$ $$\frac{(\sin2x+\sin4x)(\sin2x-\sin4x)}{\cos2x}=1$$ $$\frac{-6\sin x \sin2x}{\cos2x}=1$$ $$\tan2x+\csc6x=0$$ How to proceed after this?
Equation_Charmer
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If $\sin^2x+\sin^2y<1 \forall x,y \in R$, then prove that $\sin^{-1}(\tan x\cdot\tan y)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

If $\sin^2x+\sin^2y<1 \forall x,y \in R$, then prove that $\sin^{-1}(\tan x\cdot\tan y)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ My attempt is as follows:- $$f(x)=\tan x\tan y$$ $$f(x)=\dfrac{\sin x\sin y}{\cos x\cos y}\tag{1}$$ Let's find out…
user3290550
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Is there an exact value of $\cos^{-1}(4/5)?$

This is about the $3,4,5$ Pythagorean triangle. Question in the title. I think the answer is "no", but if not, then why not? What about the $5, 12, 13$ triangle, or even non-Pythagorean triangles like $1, 4, \sqrt {17} $ ? Can you write the angles…
Adam Rubinson
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How to prove the identity $\sin2x + \sin2y = 2\sin(x + y)\cos(x - y)$

I've been trying to prove the identity $$\sin2x + \sin2y = 2\sin(x + y)\cos(x - y).$$ So far I've used the identities based off of the compound angle formulas. I'm not quite sure if those identities would work with proving the above identity. Thank…
Ameer
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