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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Trigonometry and Quadratics

If $\tan A$ and $\tan B$ are the roots of $x^2+px+q=0$, then prove that $$\sin^2(A+B)+p \sin(A+B) \cos(A+B) + q \cos^2(A+B) = q$$ My Attempt: Using the sum and product formulae we have, $q=\tan A\tan B, $ $-p=\tan A+\tan B$ And, $\tan…
pi-π
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Different ways of solving $\sin x \cos y = -1/2$ and $\cos x \sin y = 1/2$

I was solving this question which said $$ \sin x \cos y = -1/2$$ $$\cos x \sin y = 1/2$$ and we had to solve for $x$ and $y$. One thing that I deduced was that if I simply add the two equation then I would get $$\sin x\cos y + \cos x\sin…
Harsh Sharma
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Do the series representations of the trigonometric functions depend on the definition of radian?

Suppose I were to define the notion of angle using the unit circle. By elementary geometry, I realise I could use the unit circle's area, which is an intrinsic part of the circle, as my "new" measure of angle. Thus, "a whole turn" would correspond…
Maxis Jaisi
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Cosine of average of angles

Is there an expression for the cosine of an average of two angles? I.e., If I know the cosines of $A$ and $B$, can I easily find the cosine of $(A+B)/2$? Ideally, I'm looking for something that can be computed pretty easily by hand, for instance…
brianmearns
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If $ \sin \alpha + \sin \beta = a $ and $ \cos \alpha + \cos \beta = b $ , then show that $\sin(\alpha + \beta) = \frac {2ab } { a^2 + b^2} $

I've been able to do this, but I had to calculate $ \cos (\alpha + \beta) $ first. Is there a way to do this WITHOUT calculating $\cos(\alpha+\beta)$ first ? Here's how I did it by calculating $\cos(\alpha+\beta)$ first $ a^2 + b^2 = \sin ^2 \alpha…
Nathuram
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Given ${[1+ {(1+ x)}^{1/2}]×\tan(x) = \left[1+ {(1- x)}^{1/2}\right]}$, find $\sin 4 x$.

If $${[1+ {(1+ x)}^{1/2}]×\tan(x) = \left[1+ {(1- x)}^{1/2}\right]}$$ then find the value of $\sin(4x)$. The options given are: a) $x$ b) $4x$ c) $2x$ I tried applying many trigo identities but none of them is working and the radicals are posing a…
SirXYZ
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Trigonometric equality

I would like to know, how do you simplify this: $$\cos x\sin(x+y) + \sin x\cos(x+y)$$ to this: $$\sin(2x+y).$$ Wolfram alpha says so, but how does human being do so? :)
Ondrej Sotolar
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$\min$ of expression $\sin \alpha+\sin \beta+\sin \gamma,$satisfying $\alpha+\beta+\gamma = \pi$

The $\min$ of expression $\sin \alpha+\sin \beta+\sin \gamma,$ Where $\alpha,\beta,\gamma\in \mathbb{R}$ satisfying $\alpha+\beta+\gamma = \pi$ $\bf{Options ::}$ $(a)\;\; + ve \;\;\;\;\;\;\; (b)\;\; -ve \;\;\;\;\;\; (c)\;\; 0\;\;\;\;\;\; (d)\;\;…
juantheron
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Can $ \sin(\pi - \alpha) $ be written as $\sin(180^{\circ}-\alpha)$?

This is a simple question concerning $ \sin(\pi - \alpha) $ when $ \alpha $ is known. Is it correct to write it as $$ \sin(180^{\circ} - \alpha), $$ as $ \pi $ is $ 180^{\circ} $ in radians? For example, $ \sin(\pi - 25^{\circ}) = \sin(155^{\circ})…
Deloss
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Verify identity: $\sin(x+1)\sin(x+1) - \sin(x+2)\sin x = \sin^2(1)$

I have the following identity to verify: $$\sin(x+1)\sin(x+1) - \sin(x+2)\sin x = \sin^2(1).$$ I'm becoming more familiar with sum and difference formulas to some degree, but this one has stumped me. I don't know if I'm doing it right, even, but I…
Werewoof
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How to put $\frac{1}{\cos\theta - j\sin \theta}$ in the form $a+jb$? ($j^2=-1$)

I am new to complex numbers and am having trouble putting them in the form $a+jb$ (or $a+bi$) How would I go about putting this expression in the form $a+jb$? $$\frac{1}{\cos\theta - j\sin \theta}$$
Valortome
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Interesting mathematical artifact: Equality sign wrong for exponential function.

I found an interesting case where it seems like an equality sign works wrong. Let's consider the following construction: $\frac{1+\Lambda}{2} e^{i\Lambda \phi}$ where $\Lambda = \pm1$, so $\Lambda^2=1$. Then I apply Euler…
MsTais
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How many tangent half-angle formulas are there?

\begin{align} \tan \frac{\alpha+\beta} 2 & = \frac{\sin\alpha+\sin\beta}{\cos\alpha + \cos\beta} \tag 1 \\[10pt] \tan \left( \frac \pi 4 \pm \frac \alpha 2 \right) & = \sec\alpha \pm \tan\alpha \tag 2 \\[10pt] \frac{1 +…
Michael Hardy
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Prove this trigonometry equation: $\sin 40^\circ \cdot \sin 50^\circ$ is equal to $\frac{1}{2} \cos 10^\circ$.

Prove that $\sin 40^\circ \cdot \sin 50^\circ$ is equal to $\frac{1}{2} \cos 10^\circ$. I've tried writing $\sin 40^\circ$ as $\sin(40^\circ+10^\circ)$, then wrote $\sin(50^\circ+10^\circ)$ as $\sin 40^\circ \cos 10^\circ + \cos 40^\circ \sin…
user377200
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What is $\cos[2\tan ^{-1}(x^2)]$

What is $\cos[2\tan ^{-1}(x^2)]$? I tried $$y= \tan x =\frac{\sin x}{\cos x}=\frac{\sqrt{1-\cos^2{x}}}{\cos x}$$ then $$x=\tan^{-1} y = \tan^{-1} \frac{\sqrt{1-\cos^2{x}}}{\cos x}$$ However I don't know how to deal with the $2$ in the problem to…
Matata
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