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.

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Prove the following trigonometric identity without a calculator involved

I have to prove the following statement. $$1+\cos{2\pi\over5}+\cos{4\pi\over5}+\cos{6\pi\over5}+\cos{8\pi\over5}=0$$ I have tried to use the sum of angles formula for cosine, but didn't get to a point where I'd be able to show that it is equal to…
Razvan Paraschiv
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Why is the hypotenuse in trig always positive regardless of the quadrant?

I see an image like this: In quadrant 2, even though the cosine is negative, because the x coordinate goes to the left, the hypothenuse is still positive. Why is this? It seems like the direction of x matters when determining if cos is positive or…
Jwan622
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Exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$

What are the exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$ and how do I work it out? I know that $\cos(2\pi/7)$ and $\sin(2\pi/7)$ are the real and imaginary parts of $e^{2\pi i/7}$ but I am not sure if that helps me...
Hannesh
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Extra solutions when solving $\sin\theta+\cos\theta=\sqrt{2\sin2\theta}$

The problem: $$\sin\theta+\cos\theta=\sqrt{2\sin2\theta}$$ My…
tryingtobeastoic
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Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle

Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle This question came up in a miscellaneous problem set I have been working on to refresh my memory on several topics I did earlier this year. I have…
E.O.
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Relationship between $\tanh x$ and $\arctan x$

The functions $\tanh x$ and $\arctan x$ have a similar graph. Is there a formula to transform $\tanh x$ to $\arctan x$?
Minkow
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Solve for $x$ a trigonometric equation

I want to solve for $x$ $$ {{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x $$ but I don't know how to start. Replacing $\sin x$ or $\cos x$ by $y$ led me nowhere because of the right side. One of the solutions…
Student_G
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Solving trigonometric equations of the form $a\sin x + b\cos x = c$

Suppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \cos x = 2$ where $0
rrqq
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Finding the angle between two points

First of all, I am doing some mathematical background information for a software I am creating. What I want to achieve is the point on an object rotating towards where the mouse is. Like in tank games, where the turret rotated depending on mouseX…
Moynzy
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Solve $\cos^2x-\sin^2x= 1$

I am trying to solve for $x$ in the following equation: $$\cos^2x-\sin^2x= 1$$ Given that $\cos^2x+\sin^2x= 1$, is this something I could use to solve it?
Andrew
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Why is this trigonometric identity true?

Suppose $$\frac{{{{\sin }^4}(\alpha )}}{a} + \frac{{{{\cos }^4}(\alpha )}}{b} = \frac{1}{{a + b}}$$ for some $a,b\ne 0$. Why does $$\frac{{{{\sin }^8}(\alpha )}}{{{a^3}}} + \frac{{{{\cos }^8}(\alpha )}}{{{b^3}}} = \frac{1}{{{{(a + b)}^3}}}$$
Under sky
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Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

How would I verify the following double angle identity. $$ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $$ So far I have done this. $$ (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B) $$But I am not sure how to proceed.
El Bananero
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Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance $$\cos(x^2)=…
Watson
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Found an odd relationship! Could someone help me to prove or debunk it?

I finished up in hospital which typically means that one has A LOT of spare time to kill and after using electronic devices so much that it makes you sorry I flinched into doodling and and light-headedly playing with a calculator. It happend than…
user161516
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6 answers

Is there an identity for cos(ab)?

I know that there is a trig identity for $\cos(a+b)$ and an identity for $\cos(2a)$, but is there an identity for $\cos(ab)$? $\cos(a+b)=\cos a \cos b -\sin a \sin b$ $\cos(2a)=\cos^2a-\sin^2a$ $\cos(ab)=?$
TechMaster100
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