Mathematics Stack Exchange
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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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$\sum_{n=1}^{50}\arctan\left(\frac{2n}{n^4-n^2+1}\right)$

Find the value of $$\sum_{n=1}^{50}\arctan\left(\frac{2n}{n^4-n^2+1}\right)$$ $$\frac{2n}{n^4-n^2+1}=\frac{2n}{1-n^2(1-n^2)}$$ I am not able to split it into sum or difference of two $\arctan$s.Please help me.
Vinod Kumar Punia
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How do I calculate the cartesian coordinates of stars

Given the Right ascension in h m s, Declination in deg ' " and the Trigonometric parallax How can I get the cartesian (x,y,z) coordinates of a star? I'm guessing I need 3 separate formulas to get each x, y and z values.
Justin808
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Trigonometry equation, odd-function.

So I have the following equation: $\sin\left(x-\frac{\pi}{6}\right) + \cos\left(x+\frac{\pi}{4}\right)=0$ It should be solved using the fact that Sin is an odd function, I can not really get the gripp of how and what I need to do? Any sugestions?
Per
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Do trigonometric reduction formulae work for every angle?

I'm doing some basic trig exercises like if $\sin(\theta)=k$ then find $\sin(\pi+\theta)$, etc. For instance, we know that $\sin(\pi+\theta) = -\sin(\theta)$ but is the angle $\theta$ required to be $0<\theta<\frac{\pi}{2}$ for those formulae to…
Richard Smith
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How to Find the value of a trigonometric function if other very complicated trigonometric equation is given?

Q)If $\operatorname{sin}\alpha+\operatorname{cos}\alpha=\frac{\sqrt7}2$, $0 \lt \alpha \lt\frac{\pi}{6}, then \operatorname{tan}\frac{\alpha}2 $ is:(1)$\sqrt{7}-2$(2)$(\sqrt{7}-2)/3$(3)$-\sqrt{7}+2$(4)$(-\sqrt{7}+2)/3$ i did this question by two…
anni saini
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In an acute-angle triangle $ABC$, find the minimum value of $5\tan A+2\tan B+ \tan C$.

Question. In an acute-angle triangle $ABC$, what is the minimum value of $5\tan A+2\tan B+ \tan C$? I have a form : Given $5\tan A+2\tan B+\tan C$ in a $\triangle ABC$ and $A<90,B<90,C<90$. The function $f(A,B)=5\tan A+2\tan B-\tan(A+B)$ has an…
Kevin M.
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How to have idea to prove trigonometric identities

Hi to explain this better I'll take an example. I have this identity that's giving me a hard time. $$\frac{\cos^2(a)-\sin^2(b)}{\sin^2(a)\sin^2(b)} = \cot^2(a)\cot^2(b)-1$$ This is what i would…
king
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Using the compound-angle formula, determine the exact value.

I understand how to use the compound angle formula on fractions with a numerator less than the denominator such as $$\sin\left(\frac{5\pi}{12}\right) = \sin\left(\frac{\pi}{4} +\frac{\pi}{6}\right)$$ However Im having trouble when the numerator is…
Laura
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trigonometry equation $3\cos(x)^2 = \sin(x)^2$

I tried to solve this equation, but my solution is wrong and I don't understand why. the answer in the book is: $x = \pm60+180k$. my answer is: $x= \pm60+360k$. please help :) 3cos(x)^2 = sin(x)^2 3cos(x)^2 = 1 - cos(x)^2 t =…
Silas2033
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Proving tan((x + y)/2) = (sin x + sin y)/(cos x + cos y) with the angle sum and difference identities

I've recently come across this problem of proving $$ \tan \frac{x + y}{2} = \frac{\sin x + \sin y}{\cos x + \cos y} $$ Not a difficult problem, I thought. I would have rewritten the RHS using the sum-to-product identities of sine and cosine. But…
user274548
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Derive the Trigonometric Functions

How can the Sine Function be derived? Given $\angle{A}$ as input, derive a function that would give $\frac{a}{c}$ as output. $$$$ How can the Cosine Function be derived? Given $\angle{A}$ as input, derive a function that would give $\frac{b}{c}$ as…
Paul
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Proving this trig identity:$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$

$$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$$ What I've tried, $$\frac{((1+\cos\theta)+(\sin\theta))((1+\cos\theta)+(\sin\theta))}{(1+\cos\theta-\sin\theta)…
Abhishekstudent
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Two trigonometrical answers for same exercise. Which is the right one?

I'm trying to resolve a trigonometrical exercise. I have two ways to resolve it and I receive two different answers. If you could help explain me why one way is a wrong way to resolve it (without doing reference to the trigonometric functions…
Pichi Wuana
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Simplifying $\tan100^{\circ}+4\sin100^{\circ}$

The answer is $-\sqrt3$. I was wondering if this is just a coincidence? Also, is there a relation between $$\tan(100^{\circ}+20^{\circ})=\frac{\tan100^{\circ}+\tan20^{\circ}}{1-\tan100^{\circ}.\tan20^{\circ}}=-\sqrt3$$ and the given expression? Or…
Aditya Agarwal
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How to evaluate $\tan20^\circ+\tan40^\circ+\sqrt3\tan20^\circ\tan40^\circ$ using trigonometric ratios of angles $0^o, 30^o, 45^o ,60^o, 90^o$

The question is Evaluate $$\tan20^\circ+\tan40^\circ+\sqrt3\tan20^\circ\tan40^\circ$$ using trigonometric ratios of angles $0^o,\ 30^o,\ 45^o,\ 60^o,\ 90^o$ I played with this problem for a while, but I still can't figure out how to solve it. All…
H G Sur
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