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 identity: $\frac{1+\sin\alpha-\cos\alpha}{1+\sin\alpha+\cos\alpha}=\tan\frac{\alpha}{2}$

Prove identity: $$\frac{1+\sin\alpha-\cos\alpha}{1+\sin\alpha+\cos\alpha}=\tan\frac{\alpha}{2}.$$ My work this far: we take the left…
Gjekaks
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Factor the expression and use the fundamental identities to simplify $7 \sin^2 x \csc^2 x − 7 \sin^2 x$

Factor the expression and use the fundamental identities to simplify. There is more than one correct form of the answer. $$7 \sin^2 x \csc^2 x − 7 \sin^2 x$$ I'm reviewing for a test and going over my old homework, is 7 a possible solution (I'm…
TheNewGuy
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How to solve $\tan^2 \theta − 2 \sec \theta = 2$

Solve the given equation. Find all solutions of the equation (express your answer in terms of k, where k is any integer). $$\tan^2 \theta − 2 \sec \theta = 2$$ What do I do to solve for $\theta$? update: I got $cos\theta$ $= \frac {1}{3}$ and…
TheNewGuy
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When is the cosine of $\pi/n$ of a certain form?

I have a few questions concerning $\cos(\frac{\pi}{n})$. Are there characterizations for the values $n \in \mathbb{N}$, such that $\cos(\frac{\pi}{n})$ ... is an algebraic number? ... can be written in terms of square roots? ... is of the form $a+b…
Martin
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Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
Mathxx
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Easy question Find $\sin 2x$, $\cos 2x$, and $\tan 2x$

Ok so I was absent from school yesterday because long story short I had no way to get to class b/c something happened last minute. I'm pretty sure this is easy but I keep getting the wrong answer for $\tan2x$. Find $\sin 2x$, $\cos 2x$, and $\tan…
TheNewGuy
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Some help on trigonometric equation

So I have $\sin^3x = \frac 34 \sin x$. Can you expand so the answer is either $\sin x(\sin^2x +\frac 34)$ which leads to the answer $\frac 12 + 2n\pi$ or that $\sin^3x = \frac 14(3\sin x-\sin^3x) - \frac 34\sin x$ which leads to the answer $0 + 2n…
addde
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Prove the relation for cos inverse

Prove the relation $\cos^{-1}x_0=\dfrac{\sqrt {1-x^2_0}}{x_1\cdot x_2\cdot x_3\cdots \text{ ad inf.}}$ where the successive quantities $x_r$ are connected by the relation $x_{r+1}=\sqrt{\frac{1}{2}(1+x_r)}$ My…
Vinod Kumar Punia
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Trigonometric equation with sine and cosine

So the equation is $3\cos ^2t + 5\sin t = 1$ Now I have simplified this to $$3(1-\sin ^2t) + 5\sin t -1 = 0$$ which leads to $$-3\sin ^2t + 5\sin t + 2 = 0$$ Then I get $$-3t^2 + 5 t +2 = 0$$ Is this the correct way to go with this equation then use…
addde
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Trigonometry question

If $$\frac{3-\tan^2\frac{\pi}{7}}{1-\tan^2\frac{\pi}{7}}=\alpha \cos\frac{\pi}{7}.$$ If $\alpha$ is a natural number.Find $\alpha$. My attempt is: $$\frac{3-\tan^2\frac{\pi}{7}}{1-\tan^2\frac{\pi}{7}}=\alpha \cos\frac{\pi}{7}$$ convert it into…
Vinod Kumar Punia
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Show that $ \tan (A + \theta) $ can be simplified to $- \cot \theta$ as A tends to $\frac{\pi}{2}$

So far I have used the identity, $$\tan\left(\frac{\pi}{2} + \theta\right) = \frac{\tan A + \tan \theta} {1 - \tan A \tan \theta}$$ As $A \to \frac{\pi}{2}$, $\tan A \to \infty$, so my reasoning is, $\infty + \tan \theta = \infty$, which…
Jack
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Prove that $ \tan40° + \sqrt 3 =4 \sin40° $

The equality I'm trying to prove looks like that: $$ \tan40° + \sqrt 3 =4 \sin40° $$ My guess is that $\sqrt3$ can be rewritten as $\tan60°$ and I can use proved in previous exercise formula $$\tan3 \alpha = \frac{(3 - \tan^2\alpha)\tan\alpha}{1 -…
Pavel Vergeev
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How do you find the value of $f(x)$ for this trig function satisfying all values of $x$?

If $ f(x) = 3[\sin^4(\frac{3\pi}{2} - x) + \sin^4(3\pi+x)] -2[\sin^6(\frac{\pi}{2} + x) + \sin^6(5\pi-x)] $ then, for all permissible values of $x$, $f(x)$ is:- Here's how I attempted it- $ f(x) = 3[\sin^4(\frac{3\pi}{2} - x) + \sin^4(3\pi+x)]…
Abhishek Mhatre
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Calculate Length in Perspective View

I have a 4-sided plane in a perspective view. Each side is equal in length to the side across from it. Given the length of two sides and the fore-shortened length of one side, how can we solve for the other two sides? Assume the bottom side is not…
Abdulla
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Solving an infinite series containing $\arctan$

I need to compute: $$\tan\bigg(\arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{2}{9}\right)+ \arctan\left(\frac{1}{8}\right)+\arctan\left(\frac{2}{25}\right)+\arctan\left(\frac{1}{18}\right)+\ldots\bigg)$$ I proceeded as follows. The series is…
Gop
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