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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How to interpret $\cos(n\pi)^2$

I am stuck in an elementary problem which somehow makes me confused. If it is written $$\cos(n\pi)^2$$ What does this mean? Is it $(\cos{(n\pi)}) \cdot (\cos{(n\pi)})$, or $\cos{(n^2\pi^2)}$ ??? Because if it is $(\cos{(n\pi)}) \cdot…
Kadal
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Why does $\sin(t) + \cos(t)$ itself look like a sine graph?

So the other night I was randomly python scripting. I plotted $\sin(t) + \cos(t)$ vs $t$ for $t$ ranging between $0$ to $100$ with spacings of Δt = 0.1. (It is a pretty basic code...) Anyhow, the below plot is the result. Why does it look like…
innating
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If $A$ and $B$ are solutions to $7\cos\theta+4\sin\theta+5=0$, then $\cot\frac{A}{2}+\cot\frac{B}{2}=-\frac{2}{3}$

If $A$ and $B$ are the solutions to ${\displaystyle 7\cos\theta+4\sin\theta+5=0\mbox{ where }A>0,0
Anthony Macks
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Maximizing $x_1+x_2+\cdots+x_{10}$, where each $x_i$ lies between $-1$ and $1$, and $x_1^3+x_2^3+\cdots+x_{10}^3=0$

Let $x_1, x_2, \ldots, x_{10}$ be ten quantities each lying between $-1$ and $1$, and the sum of cubes of these ten quantities is zero. Find the maximum value of $x_1+x_2+\cdots+x_{10}$. I have tried substituting $x_1=\sin(y_1)$, $x_2=\sin(y_2)$,…
Rishi Shekher
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A trigonometric identity: $\frac1{\sin40 ^\circ}+\tan10^\circ=\sqrt{3}.$

My sister asked me such a trigonometric identity (her high school challenging problem): prove: $$\frac1{\sin40 ^\circ}+\tan10^\circ=\sqrt{3}.$$ I found that this is really true (surprising... with a calculator), but as an undergraduate equipped…
youknowwho
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Adding $3\cos\left(3t + \frac\pi5\right) + 4\cos\left(4t+\frac\pi8\right)$ in order to find the period of the sum

How should I begin to add together these two trigonometric functions: $$3\cos\left(3t + \frac\pi5\right) + 4\cos\left(4t+\frac\pi8\right)$$ in order that I might obtain the value of the period of their sum? I have considered expanding the two…
Vera Leighton
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Inverse Trigonometry doubt.

Suppose $\sin y=\sin 2x$, then what will be the solution for $y$? Will it be $y=2x$ or $y=n\pi-2x$ for some $n \in \mathbb{N}$?
Adienl
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Trigonometry : Height and Distance Question

From a ship sailing due South-East at the rate of 5 miles an hour, light-house is observed to be $30^0$ North of East, and after 4 hours, it is seen due North ; find the distance of the light-house from the final position of the ship My…
Sagar Rawal
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Find $\sin^5 x + \cos^5 x$

Given $$\cos \left[\sqrt{\left(\sin x + \cos x\right)\left(1 - \sin x \cos x \right)}\right] \\{}= \sqrt{\cos \left(\sin x + \cos x \right) \cos \left(1 - \sin x \cos x\right)}.$$ Find $\sin^5 x + \cos^5 x.$ My…
piteer
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Showing $\sum_{k=0}^{n+1}\binom{n+1}{k}k\cos2kx =2^n(n+1)\cos^nx\cos(n+2)x$, and the counterpart for sine

How do I show that $$\sum_{k=0}^{n+1}\binom{n+1}{k}k\cos2kx =2^n(n+1)\cos^nx\cos(n+2)x\tag{1}$$ $$\sum_{k=0}^{n+1}\binom{n+1}{k}k\sin2kx =2^n(n+1)\cos^nx\sin(n+2)x\tag{2}$$ My try: From power-reduction formula, if $n$ is odd, we have $$\cos^{n}x =…
Shuaib Lateef
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The equation $\tan x = \tan 2x \tan 4x \tan 8x$

In the question we have the equality $$\tan 6^{\circ} \tan 42^{\circ} = \tan 12^{\circ} \tan 24^{\circ}$$ which is equivalent to $$ \tan 6^{\circ} = \tan 12^{\circ} \tan 24^{\circ} \tan 48^{\circ}$$ This means that the equation $$\tan x = \tan 2x…
orangeskid
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Compound angle formula confusion

I'm working through my book, on the section about compound angle formulae. I've been made aware of the identity $\sin(A + B) \equiv \sin A\cos B + \cos A\sin B$. Next task was to replace B with -B to show $\sin(A - B) \equiv \sin A\cos B - \cos A…
PeteUK
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Maximum value of Product of Cosines

Given $$ 0 \le \alpha_1,\alpha_2, \cdots \alpha_n \le \frac{\pi}{2}$$ and $$ \cot(\alpha_1)\cot(\alpha_2)\cdots \cot(\alpha_n)=1$$ Find the Maximum Value of $$ \cos(\alpha_1)\cos(\alpha_2)\cdots \cos(\alpha_n)$$ NB: A small hint will suffice
Umesh shankar
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Solving trigonometric identity with condition.

Problem : If $\sin\theta +\sin^2\theta +\sin^3\theta=1$ Then prove $\cos^6\theta -4\cos^4\theta +8\cos^2\theta =4$ My working : As $\sin\theta +\sin^2\theta +\sin^3\theta=1 \Rightarrow \sin\theta +\sin^3\theta = \cos^2\theta$ Now the given equation…
Sachin
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Is $\cos(2\theta)=\frac{2e^{i\theta}+2e^{-i\theta}}2$ a correct application of Euler's Formula?

I know that using Euler's Formula we can write cosine like the first expression, but concerning the second expression, is it correct like that? $$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}2$$ $$\cos(2\theta)=\frac{2e^{i\theta}+2e^{-i\theta}}2$$
P_M
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