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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How prove this $\frac{1}{\cos{A}}-\frac{\sin{\frac{A}{2}}}{\sin{\frac{B}{2}}\sin{\frac{C}{2}}}=4$

in $\Delta ABC $ not An equilateral triangle,…
math110
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How to properly write an answer for trigonometric equation cos(x)=1/2

Hello I am not sure about one thing about trigonometrics equations. Exercise is to solve the equation: $\cos(x) = 1/2$ Formula to count cos(x)=a x = x0 + 2kπ or x = -x0 + 2kπ where cos(x0) = a ,k is integer number My Answers: x0= π/3 x=π/3 +…
Paweł Michoń
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Find the period of the trigonometrics functions $\sin[\cos(x)]$ and $\cos[\sin(x)]$

So I need to find the period of these two trigonometric functions: $$\cos[\sin(x)]$$ and $$\sin[\cos(x)]$$ but algebraically and without using the graph of this functions. So to be more clear, my question is: Is there any algebraic method or any…
YoussB
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Is there a way to solve $\alpha_1\sin{(x})+\beta_1\cos{(x)}+\alpha_2\sin{(2x)}+\beta_2\cos{(2x)}=c$ for $x$?

Is there any method to solve the equation above for $x$? $$\alpha_1\sin{(x})+\beta_1\cos{(x)}+\alpha_2\sin{(2x)}+\beta_2\cos{(2x)}=c$$ This would be a trivial problem if it was only the first pair or second pair of functions. Perhaps it's also a…
AKemats
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Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$

Show $\cos(x+y)\cos(x-y) - \sin(x+y)\sin(x-y) = \cos^2x - \sin^2x$ I have got as far as showing that: $\cos(x+y)\cos(x-y) = \cos^2x\cos^2y -\sin^2x\sin^2y$ and $\sin(x+y)\sin(x-y) = \sin^2x\cos^2y - \cos^2x\sin^2y$ I get stuck at…
mikoyan
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The solutions of the equation $ \sin{x} + \sin{3x} = \frac{8}{3\sqrt{3}} $ are?

I tried this: $$ \sin{x} + \sin{3x} = \frac{8}{3\sqrt{3}} $$ $$ 2\sin{2x}\cos{x} = \frac{8}{3\sqrt{3}} $$ $$ 4\sin{x}\cos{x}\cos{x} = \frac{8}{3\sqrt{3}} $$ $$ \sin{x}(1-\sin^2{x}) = \frac{2}{3\sqrt{3}} $$ Here, I tried to set $\sin x = t$ $$…
claudiubagiu
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Why does $\sin(x)$ + $\sin(x+a)$ always come out as a sine function?

How is it possible mathematically? In short, this says, $\sin(x)$ + $\sin(x+a)$ must be equal to some function like $b$ $\sin(x+c)$ somehow, but how?
user800956
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Can it happen that an object will not cast any shadow at all?

I am puzzled by a question in Trigonometry by Gelfand and Saul on p. 57. Can it happen that an object will not cast any shadow at all? When and where? You may need to know something about astronomy to answer this question. I have drawn a diagram…
mikoyan
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How to solve these two trigonometric equations?

How to solve these two trigonometric equations : $$\sin y \sin(2x+y)=0$$ $$\sin x \sin(x+2y)=0.$$ I know one set of solution will be $(0,0)$. What will be the other set ?
user85860
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Existence of $\sin\alpha=\frac{b}{\sqrt{a^2+b^2}}; \cos\alpha=\frac{a}{\sqrt{a^2+b^2}}$

Let $a,b\in\mathbb{R}$ such that $(a,b)\neq(0,0)$. Let $x$ be a real number. We make a function such that $A(x)= a\cos(x)+b\sin(x)$. I need to show the existence of a number $\alpha\in\mathbb{R}$ such that: $$\sin\alpha=\frac{b}{\sqrt{a^2+b^2}};…
CHOSM
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Simplifying $\frac{\sec x + \csc x}{1 + \tan x}$ to an expression in terms of $\sin x$

i'm having trouble getting this one started please. Simplify the first trigonometric expression by writing the simplified form in terms of the second expression. $$\frac{\sec x + \csc x}{1 + \tan x} \qquad \sin x$$ I have tried converting…
Bucephalus
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How do I write a sum of cosines as a product of sines?

I am trying to prove that $$\cos A+\cos B+\cos C=4\sin\frac A2\sin\frac B2\sin\frac C2$$ for ABC is a triangle. I tried up to the stage of $$-2\sin^2 C+2\cos\frac{180-C}2 \cos\frac{A+B}2$$ but how do I proceed from here?
Dr. NGILAZI BANDA
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If $\tan\alpha$, $\tan\beta$ are roots of $x^2+px+q=0$, evaluate: $\sin^2(\alpha+\beta)+p\sin(\alpha+\beta)\cos(\alpha+\beta)+q\cos^2(\alpha+\beta)$

Question : Knowing that $\tan\alpha$ , $\tan\beta$ are roots of the quadratic equation $x^2+px+q=0$ ; Compute the expression $\sin^2(\alpha +\beta) +p\sin(\alpha +\beta) \cos(\alpha +\beta)+q\cos^2(\alpha +\beta$) My Working : Sum of the roots are…
Sachin
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Whats the formula for the amount to scale up an image during rotation to not see the edges

I'm trying to figure out a formula... for how much a picture (rectangle) would have to be scaled up during a rotation (at any rotation amount) so that you don't see the edge of the picture in the square of the bounding box. If the bounding box is…
badweasel
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